The meaning of "The name's" phrase
Several times I heard the phrase "The name's ...". What is the difference between the phrase and "I'm ..."/"My name is". When can/should I use it?

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

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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.

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