“No functional difference - my daughter is eligible to run for President.”
Only in your make believe imagination. Unless your daughter was born under the protection of diplomatic immunity, she is a naturalized U.S. citizen. Naturalized U.S. citizens are not eligible to serve as President.
I'd say that is an "it depends" issue. If one follows the constitution and precedent, you are correct. If one ignores the constitution and precedent (which is garden variety common), then the conclusion is incorrect.