Student leaving before completing PhD - how to write about his motivations for switching in recommendation letter?

I think you probably are overthinking this somewhat. I have written a number of letters for students in similar situations--wanting to leave my department's doctoral program (sometimes after earning a masters, sometimes not) for a better one (sometimes in a different area)--and they have generally been successful in getting into excellent programs. (These students, naturally, tend to be among the best who are enrolled in our program. On the one hand, I hate to lose them, but for the students themselves, moving is probably the best choice for their careers.) In my letters, I try to give a modest amount of context for why the students are leaving, but mostly I just try to write them a conventional very strong letter.

I don't have any firsthand knowledge of what the admissions committee at a top institution would think of a student who has switched areas multiple times, but if you explain that the student wants to switch fields and that the new field has always been his real passion (and explain why he has been unable to pursue that area previously), I imagine that would go a long way. The people reading the application might still look a bit askance of somebody who has changed subject areas repeatedly, but I don't think there's much more you can do to address that point specifically. Just make the overall letter strong.

I seriously doubt that the admissions committee is going to refuse admission to somebody who already has a funded position elsewhere. They want to get the best people for their program, and if the student has a valid reason to want to move, I think that ought to be enough. Even my own (middle-ranked) department gets some applications from people who are already funded in graduate programs at other institutions, and this has never been an issue in our decision making process.

In the long term, I don't think anybody is going to look back over a student's record and be concerned that he jumped around a few times before completing a strong Ph.D. at a good institution. People transfer between programs fairly frequently. (Most people don't move, obviously, but movers are not rare either.) And again, moving should not hurt his admission chances right now, provided he (and you) give a good explanation of why he wants to move.

So I think the most important thing you can do is to write a strong letter. Emphasize that the student already had demonstrated skill in research working with you and that you believe he has excellent potential in the new area he wants to work in. Explain why the student wants to move and that you support the decision, but don't let that be the main focus of your letter. That's just there to assure the admissions committee that the student doesn't have some problem. The core should be your strong evaluation of the student, just like in any strong recommendation letter.

Credit to you for helping your student get to where he wants to be.

Toward questions 1 and 3, without knowing more specifics (which I'm not asking you to provide, by the way), the student will need to quash some red flags in his application package. Switching fields, in and of itself, is not a red flag. Switching fields and looking indecisive is. Switching 3 times is. The appearance of a preponderance to not finish what you started is. If I were serving on an admissions committee, those would be my concerns for a student with that package.

There seem to be many factors that alleviate these concerns -- the most important to me would be that he has already been working with the group he would like to join. Obviously, your letter should work toward alleviating the committee's concerns.

Negatives are harder to discern, but if the student is right on the edge of earning the Ph.D where he is now, switching would look like odd behavior. If he has low grades on his transcript that you don't know about, that would probably torpedo his application.

if he's just at the end of his first year, he hasn't really switched twice already -- he's just had some false starts at being placed in a group, which is fairly common. You should entirely ignore this since, if you write a letter with the idea that he's repeatedly switched in your mind, it will come through between the lines.