Got it and I got it

'Got it,' 'I got that' and (especially in the US) 'I got it' are all varied forms, the first involving conversational deletion, of 'Understood' or 'I've taken that on board.' This has been covered before and can be easily checked in a dictionary. But the other sense of 'I got it' / 'Got it' (I'd agree with @Dan that they're largely synonymous in both senses) mentioned in the question is less easily to find in dictionaries. We'll look at that.

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This isn't a source I'd normally consider, but as there seems to be a lack of coverage of this reasonably common usage in the usual respected online dictionaries. PhraseMix has:

• I've got it. When you're deciding who will pay for a meal, the phrase "I've got it" means "I'll pay for the meal":

A: How much is it?

B: Don't worry about it. I've got it.

In a more general sense, you say "I've got it" when someone is offering to help with something, but you want to show that you can handle the situation without any help. For example, if you're carrying a heavy-looking suitcase and someone offers to carry it for you, you say:

That's all right, I've got it.

You might also hear:

I got it.

This isn't technically right, but it's common in American English.

I won't comment on the acceptability of the abbreviated version I got it. But it is broadened in use, being used as a paraphrase of 'You can rely on me to do that.'