Holding Position and Tendonitis
I would like to add my two cents to this discussion. My first comment is that everyone's hands and relative finger lengths are different, and one has to be careful making definitive statements about "correct" positions. I think it is always better to try and focus attention on goals (efficient fingering) and offer possible approaches where there are difficulties.
For example I have seen excellent left thumb techniques with both curved and straight joints. I have found students with tight joints (in either configuration) helped by the change to the other. I think the most relevant comment is that when playing near the tip of the thumb the least movement at the base of the thumb is necessary to operate the key. That might be achieved with a straight or curved digit.
There has been a lot of talk about cocking the left wrist but one of my favorite photos is of Taffanel with what appears to be a very straight wrist. I often think that people cock that wrist at least partly to reach an in-line open G key - perhaps an offset G would be a better option. I have a "facilitator" - my "C# finger button" (what we put our first left finger on) is extended down the flute a bit so there is not the stretch between the first two fingers. I find this incredibly relaxing. Bick Brannen designed a little plastic clip that makes this option available to anyone to try. Incidentally he has also designed plugs that fit into open holes and have covers of three varying extensions. By plugging one of these into an open G key for example you can experiment to see whether extending the ring finger less relaxes the hand.
On the question of finger position the best advice I ever got was the observation that the finger is designed for gripping motions not backward lifting. If you think that the relaxed position of the finger is just above the key (or even touching the key) the only necessary movement is to depress the finger with enough strength to work against the spring and close the key. You don't even have to lift it off. If you stop pressing the spring will push your finger up!
Needless to say a great number of successful flutists play open holed flutes without any adverse effects.
Tendonitis is not something that is suddenly just there. There are many warning signals available to the person who is observant, though in my experience few of us heed them. It sounds to me like you have a caring teacher to guide you - just be sure to pay attention to any aches and pains and keep your teacher in the picture. The most likely problems in the left hand come from stretching to cover an in-line G key. If you feel this might be the case try plugging that key and play on the rim for a while. If it helps, perhaps your next flute could have an offset G. In the right hand the most likely problem comes from the stretch between the second and fourth fingers which often occurs when the third finger is somewhat longer. If you look at your hand with fingers spread you will see that as you close it to make a fist the finger tips come together. When some hands are curved enough to bring the middle finger tip to the key the other two fingers have to be stretched sideways too much to easily cover their holes. If that is a problem, at least see that your hand is as open as possible, and that you grip as little as possible. Gripping, like making a fist, brings those second and fourth fingers sideways off their holes. Just when you want that low C to sound it is the extra effort that actually is counter-productive. In making my comments to the list it was not to suggest that everyone switch to closed holes - just the hope that some people might be able to accept it as an option without losing face (or something like that) - good luck!