TORT means wrong. It’s used with avoir in french, and be in english

Parfois, on voudrait avoir tort.

Sometimes, you just want to be wrong.

TORD means bend (verb TORDRE in conjugated form)

Il peut tordre une barre de fer de ses mains.

He can bend an iron rod with his hands.

So in your example :

" Est-ce que j’ai tort ? " is right but " Oui, tu as tort " is the right answer. Your friend is wrong.

There was some early overlapping use as a feminine adjective mentioned in Dictionnaires d’autrefois (jambe/bouche/gueule torte = jambe/bouche/gueule tordue), but the only vaguely “logical” modern connection that I can find between the notions of “tort” and “tord[…]” is in the French-Canadian expression …
“se mettre les doigts dans le tordeur” …
which, according to Pierre DesRuisseaux in Dictionnaire des expressions québécoise (I have a copy, but can’t find an online version), means “to adopt an untenable [=wrong?] position or point of view” (literally “to put one’s own fingers in/through the wringer”).

However, in spite of this early adjectival connection and my lame attempt to concoct a modern one, I agree with @Chris’ answer that your correspondent is simply wrong and I think you could respond by telling them that if they’re claiming that “tort” and “tord” are interchangeable as two officially recognized spellings of the same word, then they’re perhaps not just mangling their own fingers but the French language as well!