"Souvenir" and "memento" are almost synonyms, but "souvenir" is strongly connected with the idea of a trip or a vacation. Tourist destinations have "souvenir shops," never "memento shops." "Memento" is connected with the idea of remembered feelings or emotions. It is used for small personal gifts, or for little items taken from a place (rather than bought). "Let me give you a lock of my hair as a memento." "When they tore my old schoolhouse down, I took a little piece of wood as a memento." "Reminder" does not work here. A "reminder" is something that you use to remember something you need to do in the future. It is emotionless. It's a simple fact. "Two days before the book is due, the library sends me a reminder in my email." Oddly, instead of the word "memento" we often use the long idiomatic phrase "something to remember X by." In your sentence about the teacher, I think the best single word is "memento," but I personally would be more likely to say: "The best present for the teacher is a photo because it's something to remember the students by." P.S. Thank you, thank you, thank you for spelling "memento" correctly. Native speakers very often spell it incorrectly. I don't think I want to put the incorrect spelling into your head, but the point is that the word is connected to "memory" and has nothing at all to do with "moment."

The best thing for saying what you bought from the city or country is always "souvenir". Maybe there is another word for this but "souvenir" is the best and common. Don't try to use the word that doesn't use by native people.

Dear, as a replacement for the word souvenir, you could also use the word "remembrance". I don't suggest you using the word "reminder" because it means that there is/are thing/s that you need to be reminded. And you can use the word remembrance/souvenir even if you didn't get it from a trip. :)