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Hi, and welcome to JA!
That's a great question!
In most cases, the "First" means that it was the first Presbyterian church to be built in that town. There are "Second Presbyterian" churches, which were the second to be built, etc. (Though, typically, only the first and second ones get "numbered".)
Here is an example in Auburn, New York: http://www.westminsterauburn.org/yourti108880.html
I hope this answers your question,