We don't miss out the pronoun 'I' in the present simple tense. The sentence 'Remember that idiomatic language is a very precise thing.' can only be an imperative. If we say this we are instructing or advising another person or people to remember this fact. This sentence cannot mean 'I remember..'.

The first sentence is actually an imperative sentence ie. "You should remember that...". If a sentence doesn't have a pronoun, by the rules of English, it's an imperative. Of course, native speakers don't always follow the rules of English. There are two 'rules' I can think of that describe how pronouns are omitted in slang: 1) If you're asking a sentence in the second person present continuous (eg. "Are you going outside?"), it's common to omit "Are" or even "Are you" e.g. "You going outside?" or "Going outside?". 2) If you're asking a question in the second person present simple, either "Have" or "Have you" can be omitted e.g. "You eaten anything?" or even just "Eaten anything?" instead of "Have you eaten anything?". Keep in mind that this is VERY informal, and it only happens in very specific cases (both are questions in the second person). There may be other 'rules': the two I listed above are just ones I've noticed. As for "Got it" instead of "I've got it" (not "I got it"), that's just an idiom. Idioms don't have to make sense grammatically (just like at "Long time no see"!). This probably wasn't what you wanted to hear, but there's no rule in English that tells you how to arrive at "Got it" from "I've got it". It's just an idiom you should memorise.