As I understand it, Gondor is the big country where a lot of important stuff happens.
Minas Tirith, with its series of walls going in circles, is simply the main city of Gondor. The capital if you will.
So why does Théoden say things which suggest that the city is called "Gondor"?
Arise, arise, Riders of Theoden! Fell deeds awake: fire and slaughter! Spear shall be shaken, shield be splintered, a sword-day, a red day, ere the sun rises! Ride now, ride now! Ride to Gondor!
But... they are right there? They already are in Gondor. He should be saying: "To Minas Tirith! Attack!" or something like that.
The multiple different names for everything and everyone has repeatedly confused me throughout the books, as well as in Silmarillion, and led to an enormous amount of confusion in my head. However, I'm pretty sure that Minas Tirith is the city and Gondor is the country where the former is located. They aren't synonyms or words used by different peoples/species...
It would've made sense if he had shouted: "For Gondor!", "Ride for Gondor!" or "Fight for Gondor!", because that implies/says that they are coming to Gondor's rescue/aid, as in the nation/people, by defending their main city Minas Tirith. I was frustrated that he did not.
The "Ride to Gondor!" part partially ruins the rest of his speech for me, but perhaps this is another case of archaic English being misinterpreted by me...