No point in concealing the fact that oil and gas exports and petroleum price play important role in the Russian economy. However there is no point in highlighting this fact too much since it borders with spreading the myth that Russian economy is only oil and gas and nothing or almost nothing else.
Few facts:
Significant growth of oil prices started only in 2004-2005, but the Russian economy started growth (and rather large growth) well in 1999-2000. 
Russian economy is not only export. The recovery of oil and gas and other minerals was only 9% of GDP in 2011  and only 6.8% was oil and gas  (with figures not much larger in the previous years). The share of oil and gas in GDP gradually falls.
Unprocessed oil and gas takes only 45% of Russian exports. More of the rest are oil-based products and fuels, but this is considered manufacturing . So the phrase in one of the last edits about the "oil and gas, which is the majority of Russian exports" is oversimplified, and certainly not correct if we speak about raw materials vs manufacturing.
Federal budget grew much faster in the last decade than oil prices . And it is not clear whether "a five-fold increase in price of oil and gas" is correct description - it was four-fold increase in 2000-2011, but seven times increase in 1999-2011.
In 2008 only 23% of federal budget revenues came from exports oil and gas (customs tariffs), and 18% from taxes on petroleum and mining companies . However, when we speak about the consolidated budget (federal budget + regional budgets) the similar figures for oil and gas would be just 14% and 10% .
All this means the following:
The economic growth in the early 2000s was not very much dependent on oil and gas prices. So it is incorrect to ascribe early economic growth under Putin to petroleum prices.  
Overall, the dependence (whatever its scale) of the Russian economy on oil and gas is not Putin's invention, and the share of this sector in the Russian GDP actually is falling.
However, oil and gas money constituted a very large share of the federal (but not so consolidated) budget in the Putin era. This means that it is more correct to speak that it is not the Russian economy on the whole, but the federal budget which is dependent on oil and gas.
And a qualitative addition to the figures: Russia is a northern cold country and Russia is industrial country, which means that it needs lots of fuel for itself, but it is also a (very) resource-rich country, so no wonder that it produces much energy, have lots of facilities for that, and takes the opportunity to export abroad a lot (every country in similar conditions does that: the U.S. , Canada, Norway). This is not good or bad feature of the economy, just the fact.  Talk 
We need to revert your additions on oil and gas simply because they are not entirely correct factually, and because all the key information in more generalized and accurate form is already in the lead. But if you have proposals to make some important additions in accurate wording, let's discuss them.  Talk