"Hi, I'm Cathy" / "Hi, my name is Cathy". "I'm Cathy" is just more casual or informal. I use both methods

I'd say "The name's ____" is for formal situations, or any case where your name isn't a personal issue. The expression is impersonal by nature. So, situations would include checking your name in a registry, or even confirming that your hotel or restaurant reservation is under your name. You would also use "The name's ____" to correct someone in a formal manner.

Note that "the name's" is short for "the name is"