There is no correct answer for your example. I started living here two months ago. I have lived here for two months. (WITHOUT the word 'ago') 'ago' indicates a finished event/action and present perfect (have lived) in this sentence indicates an action that started in the past and is still true, so you cannot use 'ago." With the present perfect, "since' is used with a point in time (since yesterday, since 1:00, since last week) 'for' is used for duration (for a minute, for ten years, for a while)

In the USA, you would never hear a native speaker say "since two months ago." One would say, "I've lived here since April" or "I've lived here for two months."

"I have lived here since two months ago." The first sentence, or simply "I have lived here for two months," would be better, though.