My understanding is that lately is used for recurring events, as you mentioned, or for negative sentences with single or recurring events. I haven't seen Susan lately. Have you heard from her? Recently, only the other hand, can be used in all of these contexts.

I think your understanding is correct. And in your first sentence, you could remove "a lot" because "lately" would still imply being ill quite often in the recent past.

Well, if you translate it into indonesia, it quiet different.. But i don't know about the other language.. Lately : akhir-akhir ini Recently : belakangan ini And my preseption same as yours, that lately, is for something that you repeated again and again.. And recently, it doesnt have a spesific number, but you ever did it in the near past.. Like, ever seen and saw..