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Looked into it and I found that the Catholic temple they have today is a recent build (1895) and the Reformed is older one. They treat as evident that this church is standing where the old Catholic was (what's more, with a monastery). The old temple was destroyed several times and new was built or the old refitted, what's there today in it's form also fairly new, previously the/a tower stood by itself as a separate building.
So generally people think it was how you wrote.

When I mentioned structure I meant the town's structure I'm not sure this is clear.
Btw the Reformed church had an outer defensive wall (that redish wall with the little tower), for a time it was fashionable in certain places of the old Hungary to fortify churches.