Balcerzak & Balcerski: Surname variations

In doing family research, I've come across what I think are name variations. Are the surnames Balcerzak & Balcerski the same?
eg In a marriage registry the grooms name is listed as Balcerzak in 1814 and in 1816 at the birth of a chils he is listed as Balcerski.--I assume it is the same person as ages are okay and the name & age of the bride/mother are the same.

How about Olkoski & Olkowski? Same type of record entries.

Balcerzak may have felt Balcerzak was too common and Balcerski sounded more upscale and high tone (many noble names end in -ski), so he had it changed.
Olkoski is probably a phonetic misspelling by a semi-literate priest or scribe.

Thanks, much for your speedy answer. I have other questions. I've been translating "w obecnosci" as together with and "Urodzony Jozef Kolakowski" as the well-born" and "Dziedzic czesci" as heir to an estate. Are there better translations for these terms? Barbara