ok so i have gotten like 7 PMs about letters of interest/intent/updates, so i'll just write a few words here about my limited experience with them.
1) If the question is "should I write an LOI?", then the answer is "Why not?" Just write one. It takes like 20 minutes and might have a real impact on the the trajectory of your life.
2) Everyone's doing it now, so if you send some generic letter then it's not going to help you much at all.
2.5) A lot of times people get waitlisted because it doesn't seem like they'd matriculate if they were accepted. Schools love to keep their yield rate very high, bc theyre petty and silly like that. A gushing, GENUINE love letter is just what they are looking for to put you into the acceptance pile.
3) Write about why you loved the school, and why you are a perfect fit (BE SPECIFIC!!!)
4) Keep it under a page. Some people say a paragraph, but that's more for an update letter. I chose to combine update letters with LOI, so a page is reasonable.
5) For me, I know that the two schools I was accepted to (WUSTL and Vandy) besides my state school, specifically cited my LOIs when they called me and told me of my acceptance, so I know that they had an impact there. I also am pretty confident, because of the concordance in timing, that interest letters got me IIs at Brown, Einstein, Dartmouth, WUSTL, and perhaps others.
6) Things I talked about included: personal/family connections to the city, interest in working with a specific researcher / doing a specific, unique combined degree, and connections/experiences I had on interview day or in other interactions with the school.
7) there's no difference between letter of intent vs interest. everyone knows theyre nonbinding and that premeds will trade their pride in for that prestige. unless you have a real good reason, like ur grand-kitten's got cancer and you need to stay in whatever city Princeton Med is in in order to tend to her, then your claim that you will DEFINITELY attend if admitted is clearly bullsht.