In the Name of God, the Compassionate, the Merciful. 
Hi!
A: What happened?
B: I cut myself. ←
A: You cut yourself. Oh, but how? With a knife?
B: No, not with a knife; I fell down.
.
.
A: Oh, dear, dear, you should be more careful.
B: Yes, I know I should be more careful, but I wasn’t careful. So I fell down the stairs and I hit my head on the floor.
What does “cut” mean here"?
What is the best synonym for “cut” here?
Is “injure” the best synonym for “cut” here? ←
like this: I injured myself. ←
Thank you
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Yes, in this context, “cut” means to make a wound in the skin, usually with something sharp. However, since the speaker later clarifies that they fell down rather than using a knife or another sharp object, “injure” would be a more general synonym that fits better.
So, “I injured myself” is an appropriate synonym for “I cut myself” in this case.
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Thank you so much, Torsten 
Very nice.
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Injury is a general term for many things. It can mean a cut, broken bone, sprain, etc.
‘Injury’ is fine to use in that sentence. But it does not give much information about what type of injury. ‘Cut’ is usually used when it’s done with something sharp like a knife.
I don’t know of a word that means to open the skin and bleed caused by something that is not sharp.
I think ‘cut’ is OK in this sentence, because I can’t think of a better word. But the sentence could be made more clear. It’s fine to say:
I fell down the stairs and cut my head open.
This is a common way of saying it.
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Yes, “cut” is kind of specific as you are cutting the skin. It could also be caused by a piece of sharp metal, a sharp rock or, in my case, sharp stalks of very tough grass. In an office setting you might get a very painful paper cut.
If it is just a surface cut, you might say “it’s only a scratch.”
A very common injury from falling, is a “scrape” or “abrasion”. This is where a hand, knee, or other body part is pushed along something like concrete or rocks, removing some skin. Children are constantly scraping their knees.
If you stab yourself with something like a nail or a sharp pencil, you would have a stab wound or a puncture.
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Thank you so much, NearlyNapping 
Very nice.
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Thank you so much, Arinker 
Very nice.
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Cut can be a noun or verb. It is more common as a verb.
As a verb, slice is another word that means cut. But they are not always used in the same way. For example you might use a knife to cut your meat, but slice your bread. You use a knife for both, and the action is the same, but we use different words.
If it’s a turkey, you carve a turkey.
I don’t know if this kind of information is helpful, or if it’s too much information and confusing.
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Thank you so much, NearlyNapping 
Very nice explanation.
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