Can this sentence be translated as "May I speak"? Or is it literally asking "Am I able to speak"?
it can be understood as both. But if u want to translate those in hindi specifically with distiction, there IS an option.
Do you happen to know how to say those sentences with a distinction between them, as you mentioned?
To expressely denote asking for permission, you can use "do I have permission to speak" by way of saying "kya mujhe bolne ki ijazad hai?"
But इजाज़द is an Urdu word and would be more apt to say, अनुमती . But these days, इजाज़द is more widely used in common language and the use of अनुमती is reduced to only formal use.