OP, first it should be noted that in the form you set out his argument to be it would, strictly speaking, not be a circular argument. Saying that "P is true. Q is true. Therefore P is true" is not "circular", it is, on the other hand, tautologous. It is of the form
((P&Q)->(P)) which is always true. This is not what Descartes was saying, nor how he argued it. What he was saying was that in order for there to be any thinking or experiencing in general, there must some thing that is doing the thinking, acting or what have you. He said that existing is a necessary condition for thinking, so he would actually be arguing more closely to modus ponens: (P&(P->Q)->Q). "I know that I think, and thinking implies being, therefore, I must exist". His argument most certainly could not be "I am therefore I think" for then you would be saying "For any object, if that object exists, then that object thinks" and in order to disprove that all I have to do is say the contrary "There exists at least one object which is and does not think" and this, of course, is quite easy to show: Look around you. (Not to mention if you admit platonic entities like triangles, surely they do not "think").
The problem with your "unicorn" example is one of modality. That is to say "being pink" is only a contingent property of a unicorn (haha). Descartes argued that necessarily being precludes thinking, wheras being pink is certainly not a necessary condition for being a unicorn. Unless you are assuming that it is, but even then you would have to say "All Unicorns are Pink. Unicorns exist. Therefore there are pink unicorns". Also, also unicorns could very well be pink even if they don't exist. Depending on what you mean by exist, but then we get into some sticyk situations. (Remember the King of France? That bald guy?). I could say "How could Luke Skywalker be from Tatooine when both don't exist?", but it depends on what we mean by exists and how we judge the truth conditions of a proposition like that.
As for "I think" being able to be replaced by anything. Maybe. You could say you are deceived about everything, but thinking is a precondition for anything and is thus more fundamental, but that may just be a Western bias. The idea that "mind" is more pure and more fundamental and that we are thinking beings above all else is not necessary to the argument. You could say, most generally, I experience, therefore I am. For to experience anything must imply some form of existance even if the experience itself is an illusion.
Yes, it may be difficult to prove existence through words alone. "Don't mistake the finger for the moon". This is why, as someone pointed out, (well notneccessarily why) asian cultures focus on cultivating a type of "orginal experience" of the world, a non-judgemental form of being, through practices such as yoga and mediation and what not.
Oh and Philosophy is certainly not a part of linguistics. Linguistics is the study of language; its syntax,semantics, etc etc and justl ike any good science it found its roots growing out of philosophy.