Hi, there.

How do you imagine you work a two-button typewriter?

I imagine two possible avenues:

(1) I imagine it not working as a standard typewriter (each key press printing a character), but as a Braille typewriter: each key press produces an element of a character, and only when you are ready it advances to the next one.
Additionally, each key is connected to more than one element. As such, you shift the key forward, back, left or right, or leave it at the centre, and press. Given two five-position keys, and given letters that only use one key, you get (6*6)-1=35 possible letters. (Minus 1 because you don't have a letter that uses the 0+0 position)

(2) It can work as a simple two-button typewriter, without positions, if it writes in a trinary code, like Morse code (dots, slashes and spaces).

So, what do you think?

3579721
I always imagined something like (1), with them as being kind of like joysticks or gearshifts, actually, where half the characters are in a circle around each key, and there are cams or barriers or something that only let them move in discrete directions for each letter or punctuation mark. Each key might be able to move in about a dozen directions (say 3 each for up, down, left, and right), for 24 characters total, plus any involving shift keys on the sides of the typewriter or back-leg pedals.
It'd feel a bit like using a rotary telephone, maybe, waiting for the key to return to the center after making the movement for each character.

3579721 Morse code

It would work very well like this:

http://www.fimfiction.net/blog/212425/typewriters-in-equestria

Yes I wrote a whole blog on this last year.

3579800
Interesting, but I believe that your logic is flawed in two ways:

1)
We humans developed writing around the tentacle-like dexterity of our hands, so we can afford to have heinous degrees of subtlety on our written ideograms, like B vs ß, c vs ç vs z, u vs µ, œ vs æ, d vs ∂, n vs m vs π, etc. Never mind the five written accents in Western European languages.

Equestrians developed writing around the little fact that two thirds of their population have to write with their lips. While I'm sure that Equestrians could develop Gothic writing with a lifetime of practice, it is a fact that your average Equestrian will be writing pure chicken scratch and, as such, the writing system needs to be blunt if you hope to transmit any useful information at all. Forget about dotting the "I"s and crossing the "T"s: most people wouldn't have the dexterity to write p, d, b and q distinctly.

Typewriting didn't develop distinctly: it developed to mimic manual writing.

2)
The absolutely horrible mechanical complexity of a two-joystick 8*8*2 system, supposedly powered by the key press itself. It would be believable if it was externally powered (like a mechanical calculator), but would still be only as fast as the internal gearing can select the next key, the system can reset after each key press, and then how often you need to pause to pump the machine to keep the spring powered.

And never mind that typewriters need to be durable. Such subtle gear work may be horribly delicate, not fit for writing entire books.