Yes, both are often confused. Many people, me included, distinguish them by using "kefir mushroom" and "kefir drink" (which is the product of milk + the mushroom). I had the option to buy pre-made kefir drink but decided to go the harder road of taking care of it and feeding it myself. Gradually found my own best practices and now I get 2L of it every 4 days. (Rough dosage is a handful of mushrooms for 1L of milk; something like 15 grams or so.)

Just 2-3 sips make me feel like no coffee ever could. It's an instant health improver. And this is coming from a jaded cynical programmer. Trust me, I am not selling anything and this is not a cult -- I just wanted better health. Kefir helped a lot.

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A quick guide on how to work with the mushroom, if you need one:

You put it in a well-closed container with milk and you could open it up to "breathe" 1-2 times for several hours during the entire fermenting period (for me it's 4 full days and I open it once for 5-6h). You stir it every 12 or 24h (for me the latter). You'll quickly see why -- the water should fall to the bottom of the container and you get this slimy substance on top. NEVER USE METAL CUTLERY OR METAL STRAINER OR HOT WATER WITH KEFIR! I use wooden spoons to stir and sip it but plastic works just fine.

Some people store the stuff in a fridge but that makes it way too slow to ferment (2 weeks I've been told). I keep both of my jars in a normal room.

After the fermenting period is over, filter the product through a non-metal strainer (I use a plastic one). I opt to stir the mushroom + some thick milk left on the top of the strainer until any leftover product drains fully through the strainer. Put the product in a bottle (or any closed container). Then let the product (the fermented milk you filtered) sit in a fridge for ~12h or so (but it's still ready to drink right away, it just ferments a little more when left alone without the mushrooms inside it and my experimentation has led me to believe that makes it stronger; other people on the net say the same).

Now wash the mushrooms with cold water, thoroughly (I simply wash the mushrooms with a wooden spoon while cold tap water runs on top of it; do it while it's still in the strainer, makes it much easier to wash since any leftover water + milk gets drained and isn't getting in the way). Put the mushrooms back in your container (which you should also wash, with cold water), put new fresh milk, and you're done.

Every now and then the mushroom gets too big. You can put away some of it and freeze it in a small container with some milk (just enough to cover it, doesn't have to be 1L, I use several very small plastic jars akin to those used for blood or sperm donations). You can later unfreeze them (don't force it; leave it unfreeze by itself which could take 12-24h easily), wash it well with cold water and you can use it. Can't advise you on when and how -- you'll find out for yourself. A good rule of thumb is that it's better to risk having too much mushrooms in the milk and not too few (because that will basically net you a mediocre yoghurt and nothing else).

I have 2x 1.5L jars and my fermenting period is 4 full days. Usually takes me 15 minutes to do the whole thing I described.

Hope that helps. If you have any doubts about your gut health at all (and as a modern human in the 21st century I think we all should) then give homemade kefir a fair try. It substantially improved my life.