I think a notation that covers MOST (certainly not all) fingertricks is possible, but the more complete you make it, the less precise it will be.
First of all, define a standardized grip system: I think defining which face each thumb is resting (or which sticker, if you want to be extra precise) is sufficient. The other fingers freely float, though you could define which fingers hold the other side of the cube for completeness.
I have proposed a system of ordered pairs in which the face each thumb rests on is represented by a number: 0 = D, 1 = F, 2 = U, etc. The reason for using numbers rather than simply the letters is because you can have grips like [1, 3] and [1,-1], in which the right thumb is on different faces, but the wrist is rotated 360 degrees.
Next you define the X axis moves: R, L, l, r, and x. These moves are all done with the wrists (unless otherwise stated, but we'll get to that later). Wristing these turns results in a change to the grip. For example, starting in [1, 0] grip and performing a wrist R turn, you end up in [1, 1] grip.
An X on its own stands for a cube rotation with both wrists (e.g., going from a [0, 0] to a [1, 1] grip). To define grip changes, you use the same ordered pair system: a [0, +1] grip shift moves your right wrist clockwise one place (similar to doing an R move). Y and Z rotations get much more difficult to define concisely, so for now I'm going to just leave them alone. In general, a rotation along the same axis as the face where the thumb is is done with a "slide", and one that goes perpendicular to the thumb (for example, a y rotation from [1, 1] grip) needs a regrip of that hand. This certainly doesn't cover all cases but it's close enough for now.
Any moves that are not performed with your wrist are described with the letter of the move, then (within brackets to signify a fingertrick), a letter signifying the finger used (T-I-M-R-P for thumb index middle ring pinky, lower case t-i-m-r-p for left hand), then (optionally) a sticker where the finger begins the move, and (again, optionally), either "pull" or "push".
Thus, starting from a 1,1 grip, the move written:
U[I, BRU, pull] is the simple pull of the right index finger to perform a U. However, if you're performing the move from the sticker opposite the thumb of that hand, you can omit the starting sticker. By "sticker opposite the thumb", I mean the usual resting place for an index finger: BRU for U from [1, 1] grip, UFR for F from [1, 0] grip, etc. This applies for similar moves with the ring/pinky fingers: BRD for D from [1, 1] grip, as well as things like E and S slice turns, I suppose. Similarly, assume most moves are "pull" moves, unless specified otherwise. I think allowing for "optional" notation keeps things more concise for the majority of written algs.
Note that in my system, a "U2" generally signifies a single move. For the "double trigger" fingertrick, you'd generally write U U[M]
As an example, here are two different ways to perform a T-perm:
[1, 1] R U[I, BRU] R' U' R' F R2 U' R' U' R U[I, BRU] R' F'[T, DFR]
[1, 0] R U R' [0, +1] U' R' F R2 U' R' U' [0, -1] R U R' F'[T, DFR]
Anyway, I hope this makes as much sense to you all as it does to me. This system allows you to flexibly describe most fingertricks in use. Granted, kirjava's example doesn't work (although I have a few algs in this modified "column" grip and I think it could be separately defined), and I'm sure there are a million other exceptions, but it mostly works. Let me know if anything needs more explaining.