The other answers don't really discuss the technical part, they just give advice on how to deal with varying opinions. This is very valuable, but I thought discussing technique would still be an important addition. (After all, that's what the question is about.)
Now I am not familiar with Dorothy Taubman, but looking at the Wikipedia article you linked to it seems I am familiar with the concepts she proposes: most of them are widely taught nowadays, she just integrated many of them in a comprehensive model. (And the 8 principles listed are hard to disagree with.)
I don't think your new teacher disagrees with Taubman's work. Not least because the concepts Taubman proposes are (nowadays) not controversial save the forearm rotation.
General technical considerations
You can play the piano using many different muscle groups. Not only the fingers and the wrists are involved, but so are the arms and shoulders and certain muscles in the back. Additionally, posture also has a considerable influence on your technique.
Usually, a distinction is made between the (very broad) categories of finger playing, hand playing and arm playing. These three are all valid playing techniques and necessary to play certain things. You almost always need a combination.
Finger playing is the easiest to teach to beginners. Although arm playing is not very difficult to teach either, finger playing immediately gives more control to the player so it is often taught first for motivational reasons. (Personally, I see no reason to pick either, you can develop both in parallel. Only combining them is difficult.) For this reason, we often see people who have very developed finger playing, but don't involve the rest of their playing apparatus.
Because beginning students often only use their fingers while playing, piano teachers often (have to) stress the wrist motion. This is compensatory since in ideal technique, you use a combination of all muscle groups, in other words, the whole playing apparatus; sometimes a bit more of this, sometimes a bit more of that, but it occurs very rarely (never say never) that you would use only one element. (Cf. also Taubman.)
If you use only your fingers, usually your wrists "lock up"; you fix them in place by flexing all kinds of surrounding muscles. All this tension is of course a hindrance in playing. Training specific wrist motions requires these muscles to relax and frees up the wrists. This is important because locked wrists cut the connection with the arm and therefore prevent use of the arm in piano playing. Hand playing is mostly a hinging motion in the wrists so if you lock those it's completely impossible.
(This is where Taubman's controversial forearm rotation comes in. She says it should be minimal and in my opinion maybe to the onlooker imperceptibly so, it just serves to keep the wrist free. If you make this rotation large enough to be clearly visible it obviously becomes a hindrance – this is why I think people take issue with it.)
What I think happened to you
For whatever reason, you avoided the trap of playing only with the fingers while locking the wrists in your self-taught piano journey. (Kudos!) However, since finger playing is mostly assumed to be overdeveloped, it never really came up in the materials you used so naturally you never really trained it. Now, your teacher sees that your arm playing is very developed but your fingers are lagging behind. She is now trying to get you to compensate for this gap in your technique.
I think this is the case because of multiple reasons:
- "most of the movement to hit a key should come from the fingers alone" nobody who plays on a professional level can seriously believe this as a general policy: in certain contexts it can be true, but please try and play a large chord with the fingers alone for example. Since the nineteenth century nobody has made serious claims that the wrists should not be involved at all.
- On the other hand, if I had to teach a pupil whose finger playing was underdeveloped (relative to the rest of their technique) this is exactly what I would tell them to do. Not because that is ideal technique, but to train the parts that are lagging behind. If your left leg is substantially stronger than your right and you are limping, would it perhaps be a good idea to do some additional strength training for the weaker leg? Or would you just keep on trying to run as close to in perfect form as you could? (Strength is part only of the analogy; I'm not saying finger strength is the problem – without seeing you play what I have said before is going out on a limb already.)
- The Russian school actually starts with and stresses arm playing, as does Taubman apparently. So the fact that your teacher was trained in the Russian tradition supports my hypothesis. Yes, the Russians also talk about "perlé" playing but that is only useful on a thorough foundation which includes a flexible wrist. ("Perlé" actually comes from the French school but it's all the rage in Russia now while the French have moved past it.)
Taubman vs. the USSR
If there is actual disagreement between Taubman and your new teacher at all, I think the most plausible difference would be the mental model used to combine the various elements of the playing apparatus.
One way to look at it is to let the fingers lead the wrists. If the wrists are free, they will follow the fingers comfortably and conveniently and support the fingers in striking the keys.
Another way to look at it is to place the fingers using an organic and effortless wrist motion. If the finger playing is developed enough they will fall into place as an extension of the wrists.
That you need both wrists and fingers is a fact. The importance of either element depends on the notes to be played, the hands you are stuck with, the sound you want to produce and personal opinion. I hope you can imagine that the two approaches above become equivalent if the importance of the two components vary independently of the mental model used.
Personally, I need both: in some passages I can't get my fingers to the right place without consciously correcting the motion of my wrist, in other passages I am hindering my fingers by excessive wrist flailing which I can only solve by relaxing the wrist and letting the fingers lead it to the right place.
In my lessons with my teacher, it has become apparent ...
This doesn't sound like you actually talked about your concerns with her. I highly recommend you do; clearly she is in a much better position to give advice about your technique than anyone on this site is, since she actually knows you and how you play.
If my above answer is right, the root cause of your problem is a lack of communication between the two of you: you think she is teaching you what ideal technique looks like, but actually she is making you fix a hole in your technique through some targeted exercise, neither realising what the other is thinking.
If you feel like you can't talk about this with her this is a separate, highly important, issue you should solve first and as soon as possible.