I think you're half right. The Constitution does not address term limits at all, so term limits MUST be legal because...
I has been a few months since I last read the Constitution, but I don't remember a Constitutional Amendment being "passed" to limit presidential service to two terms.
Can you tell me where that Constitutional Amendment falls within the Constitution?
Unless of course you want to claim that every "law" passed constitutes and amendment... even those found to be "unconstitutional."
Twilight Zone time again?
22nd Amendment. So it is a Constitutional issue. Since an amendment has to be approved by a 2/3rds majority in both the House and Senate, by definition is is veto-proof and no presidential signature is required, so it goes straight to the states for ratification. The only role for a president is to advocate pro or con.
"In May 1995, the United States Supreme Court ruled 5-4 in U.S. Term Limits, Inc. v. Thornton, that states cannot impose term limits upon their U.S. Representatives or U.S. Senators." (see: https://ballotpedia.org/Term_limits_in_the_United_States). An attempt by the 1995 Contract with America Republican Congress to pass the necessary amendment failed to meet the two thirds majority necessary.