Actually there is a situation in which "I have my haircut this morning" is correct. First, it would have to be haircut (the noun) instead of hair cut (which is an object verb pair). Then, it would have to be morning time too. In this case, the meaning would be "this morning, I have a haircut appointment" However, as written, "I have my hair cut this morning" is incorrect. My immediate reaction was "I had my hair cut this morning" as in "[it is no longer morning and] my hair was cut in the morning"

I cut my hair - means I do it myself I have my hair cut- Means someone cuts my hair for me I had my hair cut- Means I got someone cut my hair in the past

"Have" is present, and "had" is past. That said, if you are talking about one haircut, your first example would more naturally be expressed in the present using a continuous aspect -- "I am having my hair done" -- and it would generally suggest you are about to have it done, rather than something you would say as the scissors clip away. If you say "I had my hair done this morning", that would mean it is finished now. You could use "have", but it would be in the context of a continuing arrangement, e.g. "I have my hair done every two months".

You are correct. The first sentence should be "I had my hair cut this morning". It should be "had" because "this morning" tell us that this action happened in the past.

The only place online where I can find "I have my hair cut this morning" is on a Chinese website, not an English one. The writer has made a mistake. You are correct: it should read "had".