Who decides how such a historic piece is adjusted for modern instruments?
Or do players learn to adjust themselves?
Of course. That's precisely what learning to play an instrument is all about, whether the piece was written in 1621, 2021, or any other year. No two performers play the same piece identically, and indeed the same performer doesn't necessarily play the same piece identically every time.
Part of learning to play the modern piano or violin or what have you is learning how to make music with the instrument rather than just playing the notes on the page. Phrasing, articulation, and expression are different on modern instruments as opposed to older ones, but if you consider a single instrument they also differ from one style of music to the next. Beethoven sounds different on a piano built in 1790, 1830, or today, but on a modern piano one plays Beethoven differently from Bach and differently from Brahms and differently from Stravinsky. Learning to play any instrument means learning to play all different parts of its standard repertoire effectively, and with the piano the standard repertoire reaches back to the 18th century, with occasional appearances by pieces from as early as the 16th.
On a larger scale (i.e. an Orchestra) and with enough ambiguity, it seems that decisions might be made which drastically change a piece. Or that modern assumptions about a composer/period would overly-inform how a piece is played.
Overly? What's "too much" here? Our modern assumptions are what they are. Perhaps they will change over time as we learn new things (or forget old things) about the past.
But our audiences are modern, too. All we can do is read what people wrote about performance practice at the time, see what we can learn from it, and produce the best performance we can. One person's idea of "best" will differ from another's. I saw a recording a few months ago of a master class given by the pianist Murray Perahia on a Bach piece. He mentioned that the music had been written for a different instrument and spoke of the things a pianist must do to deal with that. Different pianists approach the issue differently. But, as with all performances, the choices shaping the performance are the performer's.
In a comment, you ask:
It's interesting you put way more emphasis on players than other answers. I presume the average classical musician could play from any edition in your opinion?
I emphasize the performers (not always "players") because it is the performers who make the choices that the audience hears. One such choice is to decide which edition to use if multiple editions are available. That is, the editor's decisions are little more than advice to the performer. Even after choosing an edition, the performer can (should!) choose which of the editor's marks to observe and which to disregard. That is why a good edition makes it clear which aspects of the edition are in the source and which are due to the editor, as well as the reasons behind editor's choices.
It may not even be possible to notate subtle choices such as the one mentioned in the question, articulation of the third movement of the "Moonlight" Sonata. For example, there are infinite degrees of variability in the articulation of a simple scale. One infinitely variable parameter is the relative time between the damper of one note falling on the string and the hammer of the next note hitting its string (or perhaps these things happen in the opposite order). There is also the velocity of the hammer (and of the damper) to consider, and probably others besides.
These subtle distinctions can't be captured fully with articulation marks such as the staccato dot, a legato slur, and so on; there is only a finite set of permutations of this finite set of symbols. That infinite variability is the province of the performer alone; notation (and even instructions from a conductor, teacher, or coach) can only be a guideline.