Learn Hardcore Turkish: Limonlu su yazın beni serinletiyor. - Lemon water cools me down in the summer.

Yazın is a set adverb meaning in (the) summer. Turkish usually expresses seasons-as-time with this adverbial form, not with the locative case.

Natural: yazın, kışın (in winter), baharda (in spring is also common, but ilkbaharda is clearer).
Not idiomatic for seasons: yazda.
Alternatives:
- yazları = in summers, every summer (habitual).
- yaz aylarında = in the summer months (more formal/explicit).

Is yazın the same word as the imperative yazın (Write!)? How do I tell them apart?

They’re homographs but different in function.

yazın (time adverb) = in summer.
yazın (imperative 2nd person plural/formal) = you (plural/formal), write! You tell by context. If there’s no verb that could be an imperative and yazın sits where a time adverb makes sense, it’s the seasonal adverb. Here, the verb is serinletiyor, so yazın must be “in summer.”

Why is it serinletiyor and not serinletir? Which tense is better?

Both are possible; they differ in nuance.

serinletiyor (-iyor) = present continuous/ongoing or present-in-the-current-period. With a time frame like , it often means “these days/this summer, lemon water is cooling me down (repeatedly).”