There is no "Second Presbyterian" denomination that I know of.
This most likely means that this church was the second Presbyterian church founded in that particular town. Keep in mind that there may be even more of them; perhaps a Third Presbyterian or Fourth Presbyterian, depending on how large the Presbyterian population is in the town.
Banks are notorious for names like this. Any town in the US with a population of 10,000 people probably has a First National Bank. Officially, it will probably be the "First National Bank of <Town Name>", but everyone will simply refer to it as First National Bank. However, since no one likes being #2, you rarely see a Second National Bank (or church, for that matter). Instead, they'll name it something more generic; for instance, perhaps using the name of the bank's founder.