write down - разве это не "ЗАписать"?
"напиши" здесь означает "напиши/пришли мне"

So write them down and send them to me?

Думаю, down здесь лишнее.

Я буду тебе писать (письма)
Пиши (мне) почаще
Напиши (мне), как у тебя дела
Напиши (мне), какя у вас погода
Напиши (мне), как ты себя чувствуешь и какие лекарства принимаешь

let me know all the medications you're (currently) taking (?)

When we list things for someone, we write them down: https://www.google.com/search?q=%22write+down+all+medications%22&client=firefox-b-1-d&hl=en&sxsrf=ALeKk03-KpitrJjE5aRKxMZWK903egzP_A%3A1626011147238&ei=C_bqYLrzDZCqtQbC947QDA&oq=%22write+down+all+medications%22&gs_lcp=Cgdnd3Mtd2l6EAMyBQghEKABMgUIIRCrAjoHCCMQsAMQJzoHCAAQRxCwAzoHCC4QsAMQQzoECCMQJzoFCAAQkQI6CAgAELEDEIMBOg4ILhCxAxCDARDHARCjAjoLCC4QsQMQxwEQowI6BAgAEEM6BQguELEDOgIIADoHCAAQQxCLAzoFCAAQiwM6EQguELEDEIMBEMcBEK8BEIsDOggIABCxAxCLAzoHCAAQsQMQQzoKCAAQsQMQChCLAzoFCAAQsQM6BQguEIsDOgUIABDLAToGCAAQFhAeOgUIABCGAzoHCCEQChCgAToICCEQFhAdEB5KBAhBGABQotDEAliR88QCYMf1xAJoA3ACeACAAcABiAHWHJIBBDAuMjmYAQCgAQGqAQdnd3Mtd2l6yAEKuAECwAEB&sclient=gws-wiz&ved=0ahUKEwj6pNPGk9vxAhUQVc0KHcK7A8oQ4dUDCA0&uact=5

First, write down all medications your parent is taking - во-первых, запишите/записывайте все лекарства, которые ваши родители принимают

Write down all medications and herbal products you are taking and have them ready to bring with you. - Составьте список .... и имейте его наготове ...

It is important to write down all medications you are currently taking - Важно записать (/составить список)

И т.д. - все в том же духе. Везде действительно нужен (полный, формальный) список.

В данном же случае человеку просто интересно, от каких лекарств такой хороший результат, и просит сообщить. Если бы это был телефонный разговор, просто спросил бы, какие лекарства собеседник принимает. "Написать" в данном случае - просто способ коммуникации.

Я не настаиваю - сам решай, и Дмитрий пусть сам думает.

И даже all здесь лишнее IMO.

I agree. I just reworded it, but I don’t understand why we can’t use WRITE in both languages.

Is it because we say “write down” without “for me”? In some contexts it’s not necessary. For instance with RECOMMEND and EXPLAIN, “to me” sounds unnatural: Please recommend something I can watch with my children on Netflix. -or- I’d appreciate it if you could explain your reasons for leaving the US.

The same applies here. We say WRITE DOWN in the sense of putting it down on paper so that *I* can remember it, reference it, etc.: “I really enjoyed chatting with you. Please write down your number — I’d love to meet up again.”

To me your translation sounds like "запиши лекарства и храни список, имей его наготове" - like in most examples in google.

And I’m trying to explain that in this context, it means write them down FOR ME, we just don’t say that outright.

Look at the first paragraph under the line in red: https://sports.mynorthwest.com/838953/oneil-busting-myths-seahawks-draft/?

He makes it clear that the purpose is to eventually “mail it to me” or “text it to our show”… I just WANT a record…

Yes, for a record - hence DOWN.
And what if it wasn't followed by "mail or text" - like in this case?
I don't even understand why the name has to be written down, not just texted or e-mailed - unless it for the other person's own record.

it's

Because when you say WRITE DOWN, it tells the listener that your requesting the information, as a favor, for your reference, etc. By that logic “Write down your phone number” is another way of asking “May I have your phone number?” except that you don’t have to specify “may I have” or “for me.” In the context of this post, if I saw what good results you got from your medications and I wanted to try them myself, I’d naturally say “Hmmm, write down the names of your medications — I’ve been looking for something to help with my acne.”

OK, you know better, but for me "write it down" in this context wouldn't imply sending the information in the nearest communications. Most likely, it's just me.

👍🏼