Revert the changes in `arrow_writer.py` from #6636

#6636 broke `write_examples_on_file` and `write_batch` from the class `ArrowWriter`. I'm undoing these changes. See #6663.

Note the current implementation doesn't keep the order of the columns and the schema, thus setting a wrong schema for each column.

The docs for this PR live [here](https://moon-ci-docs.huggingface.co/docs/datasets/pr_6664). All of your documentation changes will be reflected on that endpoint. The docs are available until 30 days after the last update.

> Hi! We can't revert this as the "reverted" implementation has quadratic time complexity. Instead, let's fix it:

I agree, but it's the implementation we have had so far. Why don't we:
1. Release a hotfix ASAP (since would be doing a revert, we know it works as before) so people can continue using this library fine since AFAIU right now mostly writing examples for people is broken.
2. Then, focus on still applying the performance improvement and release again

The fix is straightforward, so one patch release (after this PR is merged) is enough.

Btw, let's also add a test to `tests/test_arrow_writer.py` to avoid this issue in the future.

> Btw, let's also add a test to tests/test_arrow_writer.py to avoid this issue in the future.

Would you mind adding such test, as you're more familiar with the codebase?

<details>
<summary>Show benchmarks</summary>

PyArrow==8.0.0

<details>
<summary>Show updated benchmarks!</summary>

### Benchmark: benchmark_array_xd.json

| metric | read_batch_formatted_as_numpy after write_array2d | read_batch_formatted_as_numpy after write_flattened_sequence | read_batch_formatted_as_numpy after write_nested_sequence | read_batch_unformated after write_array2d | read_batch_unformated after write_flattened_sequence | read_batch_unformated after write_nested_sequence | read_col_formatted_as_numpy after write_array2d | read_col_formatted_as_numpy after write_flattened_sequence | read_col_formatted_as_numpy after write_nested_sequence | read_col_unformated after write_array2d | read_col_unformated after write_flattened_sequence | read_col_unformated after write_nested_sequence | read_formatted_as_numpy after write_array2d | read_formatted_as_numpy after write_flattened_sequence | read_formatted_as_numpy after write_nested_sequence | read_unformated after write_array2d | read_unformated after write_flattened_sequence | read_unformated after write_nested_sequence | write_array2d | write_flattened_sequence | write_nested_sequence |
|--------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 0.005083 / 0.011353 (-0.006270) | 0.003697 / 0.011008 (-0.007311) | 0.063302 / 0.038508 (0.024794) | 0.028866 / 0.023109 (0.005757) | 0.249987 / 0.275898 (-0.025911) | 0.270803 / 0.323480 (-0.052677) | 0.004096 / 0.007986 (-0.003890) | 0.002752 / 0.004328 (-0.001577) | 0.049156 / 0.004250 (0.044906) | 0.042936 / 0.037052 (0.005884) | 0.266907 / 0.258489 (0.008418) | 0.291462 / 0.293841 (-0.002379) | 0.027703 / 0.128546 (-0.100844) | 0.011006 / 0.075646 (-0.064641) | 0.206238 / 0.419271 (-0.213033) | 0.035446 / 0.043533 (-0.008087) | 0.248923 / 0.255139 (-0.006216) | 0.264141 / 0.283200 (-0.019058) | 0.017545 / 0.141683 (-0.124138) | 1.157145 / 1.452155 (-0.295009) | 1.199007 / 1.492716 (-0.293710) |

### Benchmark: benchmark_getitem\_100B.json

| metric | get_batch_of\_1024\_random_rows | get_batch_of\_1024\_rows | get_first_row | get_last_row |
|--------|---|---|---|---|
| new / old (diff) | 0.092741 / 0.018006 (0.074734) | 0.299057 / 0.000490 (0.298567) | 0.000211 / 0.000200 (0.000011) | 0.000049 / 0.000054 (-0.000006) |

### Benchmark: benchmark_indices_mapping.json

| metric | select | shard | shuffle | sort | train_test_split |
|--------|---|---|---|---|---|
| new / old (diff) | 0.017936 / 0.037411 (-0.019475) | 0.061552 / 0.014526 (0.047026) | 0.072938 / 0.176557 (-0.103618) | 0.118192 / 0.737135 (-0.618944) | 0.074589 / 0.296338 (-0.221750) |

### Benchmark: benchmark_iterating.json

| metric | read 5000 | read 50000 | read_batch 50000 10 | read_batch 50000 100 | read_batch 50000 1000 | read_formatted numpy 5000 | read_formatted pandas 5000 | read_formatted tensorflow 5000 | read_formatted torch 5000 | read_formatted_batch numpy 5000 10 | read_formatted_batch numpy 5000 1000 | shuffled read 5000 | shuffled read 50000 | shuffled read_batch 50000 10 | shuffled read_batch 50000 100 | shuffled read_batch 50000 1000 | shuffled read_formatted numpy 5000 | shuffled read_formatted_batch numpy 5000 10 | shuffled read_formatted_batch numpy 5000 1000 |
|--------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 0.287186 / 0.215209 (0.071977) | 2.795694 / 2.077655 (0.718039) | 1.474386 / 1.504120 (-0.029734) | 1.359065 / 1.541195 (-0.182130) | 1.375295 / 1.468490 (-0.093196) | 0.569448 / 4.584777 (-4.015329) | 2.374428 / 3.745712 (-1.371284) | 2.770198 / 5.269862 (-2.499663) | 1.716346 / 4.565676 (-2.849330) | 0.063173 / 0.424275 (-0.361102) | 0.005031 / 0.007607 (-0.002576) | 0.333197 / 0.226044 (0.107153) | 3.271739 / 2.268929 (1.002811) | 1.826406 / 55.444624 (-53.618218) | 1.554537 / 6.876477 (-5.321939) | 1.565927 / 2.142072 (-0.576146) | 0.649796 / 4.805227 (-4.155431) | 0.118371 / 6.500664 (-6.382293) | 0.042536 / 0.075469 (-0.032933) |

### Benchmark: benchmark_map_filter.json

| metric | filter | map fast-tokenizer batched | map identity | map identity batched | map no-op batched | map no-op batched numpy | map no-op batched pandas | map no-op batched pytorch | map no-op batched tensorflow |
|--------|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 0.969882 / 1.841788 (-0.871906) | 11.638201 / 8.074308 (3.563893) | 9.759370 / 10.191392 (-0.432022) | 0.128069 / 0.680424 (-0.552355) | 0.013493 / 0.534201 (-0.520708) | 0.287324 / 0.579283 (-0.291959) | 0.267542 / 0.434364 (-0.166821) | 0.320072 / 0.540337 (-0.220265) | 0.421132 / 1.386936 (-0.965804) |

</details>
PyArrow==latest

<details>
<summary>Show updated benchmarks!</summary>

### Benchmark: benchmark_array_xd.json

| metric | read_batch_formatted_as_numpy after write_array2d | read_batch_formatted_as_numpy after write_flattened_sequence | read_batch_formatted_as_numpy after write_nested_sequence | read_batch_unformated after write_array2d | read_batch_unformated after write_flattened_sequence | read_batch_unformated after write_nested_sequence | read_col_formatted_as_numpy after write_array2d | read_col_formatted_as_numpy after write_flattened_sequence | read_col_formatted_as_numpy after write_nested_sequence | read_col_unformated after write_array2d | read_col_unformated after write_flattened_sequence | read_col_unformated after write_nested_sequence | read_formatted_as_numpy after write_array2d | read_formatted_as_numpy after write_flattened_sequence | read_formatted_as_numpy after write_nested_sequence | read_unformated after write_array2d | read_unformated after write_flattened_sequence | read_unformated after write_nested_sequence | write_array2d | write_flattened_sequence | write_nested_sequence |
|--------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 0.005679 / 0.011353 (-0.005674) | 0.003746 / 0.011008 (-0.007262) | 0.050149 / 0.038508 (0.011641) | 0.034382 / 0.023109 (0.011273) | 0.289802 / 0.275898 (0.013904) | 0.314993 / 0.323480 (-0.008487) | 0.004488 / 0.007986 (-0.003498) | 0.002786 / 0.004328 (-0.001542) | 0.047987 / 0.004250 (0.043737) | 0.046589 / 0.037052 (0.009537) | 0.301420 / 0.258489 (0.042931) | 0.335384 / 0.293841 (0.041543) | 0.050701 / 0.128546 (-0.077845) | 0.010987 / 0.075646 (-0.064660) | 0.058292 / 0.419271 (-0.360979) | 0.033973 / 0.043533 (-0.009560) | 0.288923 / 0.255139 (0.033784) | 0.306263 / 0.283200 (0.023064) | 0.018856 / 0.141683 (-0.122827) | 1.160721 / 1.452155 (-0.291433) | 1.208151 / 1.492716 (-0.284565) |

### Benchmark: benchmark_getitem\_100B.json

| metric | get_batch_of\_1024\_random_rows | get_batch_of\_1024\_rows | get_first_row | get_last_row |
|--------|---|---|---|---|
| new / old (diff) | 0.092633 / 0.018006 (0.074626) | 0.300353 / 0.000490 (0.299864) | 0.000219 / 0.000200 (0.000019) | 0.000045 / 0.000054 (-0.000009) |

### Benchmark: benchmark_indices_mapping.json

| metric | select | shard | shuffle | sort | train_test_split |
|--------|---|---|---|---|---|
| new / old (diff) | 0.022257 / 0.037411 (-0.015154) | 0.075417 / 0.014526 (0.060892) | 0.087289 / 0.176557 (-0.089268) | 0.125416 / 0.737135 (-0.611720) | 0.088751 / 0.296338 (-0.207588) |

### Benchmark: benchmark_iterating.json

| metric | read 5000 | read 50000 | read_batch 50000 10 | read_batch 50000 100 | read_batch 50000 1000 | read_formatted numpy 5000 | read_formatted pandas 5000 | read_formatted tensorflow 5000 | read_formatted torch 5000 | read_formatted_batch numpy 5000 10 | read_formatted_batch numpy 5000 1000 | shuffled read 5000 | shuffled read 50000 | shuffled read_batch 50000 10 | shuffled read_batch 50000 100 | shuffled read_batch 50000 1000 | shuffled read_formatted numpy 5000 | shuffled read_formatted_batch numpy 5000 10 | shuffled read_formatted_batch numpy 5000 1000 |
|--------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 0.286477 / 0.215209 (0.071268) | 2.801931 / 2.077655 (0.724277) | 1.553034 / 1.504120 (0.048914) | 1.426152 / 1.541195 (-0.115043) | 1.443824 / 1.468490 (-0.024666) | 0.563298 / 4.584777 (-4.021479) | 2.428968 / 3.745712 (-1.316744) | 2.685964 / 5.269862 (-2.583897) | 1.752304 / 4.565676 (-2.813372) | 0.064174 / 0.424275 (-0.360101) | 0.005079 / 0.007607 (-0.002528) | 0.344899 / 0.226044 (0.118855) | 3.372528 / 2.268929 (1.103600) | 1.900723 / 55.444624 (-53.543901) | 1.623721 / 6.876477 (-5.252756) | 1.781009 / 2.142072 (-0.361064) | 0.655229 / 4.805227 (-4.149998) | 0.116050 / 6.500664 (-6.384614) | 0.040374 / 0.075469 (-0.035095) |

### Benchmark: benchmark_map_filter.json

| metric | filter | map fast-tokenizer batched | map identity | map identity batched | map no-op batched | map no-op batched numpy | map no-op batched pandas | map no-op batched pytorch | map no-op batched tensorflow |
|--------|---|---|---|---|---|---|---|---|---|
| new / old (diff) | 1.004714 / 1.841788 (-0.837074) | 12.108179 / 8.074308 (4.033871) | 10.233447 / 10.191392 (0.042055) | 0.141438 / 0.680424 (-0.538986) | 0.015387 / 0.534201 (-0.518814) | 0.288068 / 0.579283 (-0.291216) | 0.277025 / 0.434364 (-0.157339) | 0.331714 / 0.540337 (-0.208623) | 0.424209 / 1.386936 (-0.962727) |

</details>
</details>

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