Tanks for the laugh, but the accepted definition of legislative branch is "branch of government that makes laws" and the president is very much in the part of the government that makes laws, considering the difficulty in passing them without their approval.
No, as others have already explained to you, the president is not part of the branch of government that makes laws. He is (often) part of the process, but he's not part of the branch - not as those words are normally used.
You can go with your own personal interpretation of the definition, or you can talk with the rest of the planet. Your choice.
You basically can’t make a law without the president. I can’t see how you can claim they’re not part of the law making process. Two thirds of both houses means the president absolutely is a a part of the branch of government that makes laws.
