"Maybe" is more conversational and less likely to be used in formal writing, but other than that the two words are very often interchangeable -- as they are in all your dictionary examples. There might be some expressions in which only one of "maybe" or "perhaps" is idiomatic, but none immediately come to mind.
John is not a engineer, maybe a technician.
This is not very good English. You could say John is not a engineer; maybe/perhaps he's a technician. or John is not a engineer; he may be a technician. In ordinary conversation John is not a engineer ... maybe a technician. is a kind of shorthand, but I wouldn't punctuate it with a comma.
Perhaps/maybe, but I'm not sure about that.
Fine.
Perhaps, I may have a car.
I'd punctuate this as Perhaps I may have a car (unless you actually mean Perhaps. I may have a car.). Strictly you don't need both "perhaps" and "may", but in ordinary conversation it's the kind of thing one would say. Actually, to answer my own question, when the sentence (redundantly) also includes the word "may", it would sound odd to use "maybe". For example, Maybe I may have a car is strange (while Maybe I'll have a car is fine).