Neither "I got my finger cut" nor "I had my finger cut" is correct.
"I cut my finger" is how we say it. It's normally the result of an accident but you were still involved in it. Even if you just accidentally run your finger along a sharp knife while doing the washing up, we say "I/I've cut my finger" - it doesn't have to be on purpose. If someone else attacks you with a knife, you might say "He cut my finger" or "He cut my arm" but if the attack penetrates the skin, then you would say "I've been stabbed", not "cut".
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