The question of the recording location and the nature of official vital records is actually independent of placenames and jurisdictions (and WikiTree's categorizations of same).
The changeover date for vital records in the Kingdom of Hungary is October 1, 1895: that was the day that civil registration began. Before then, church registers were considered to be official government records, and the register where any particular event was required to be recorded was determined by the place of residence and the religion of the participants. For example, my spouse's grandfather was born in Kula, Bács-Bodrog county, where the local church was Roman Catholic. His family was Lutheran, so his birth is recorded in Cservenka, the next town north, where there was a Lutheran church. There was also a Lutheran church in the next town south, Verbász, but that's not where Lutherans of Kula were obligated to go. (All three places are now in Serbia.)
Jewish records before civil registration are a thorny topic: rampant anti-Semitism combined with rules-lawyering meant that people in power could -- and did -- deny the acceptability of Hebrew records, resulting in situations like what you described, where a marriage's validity was called into question.
(And then there's the other extreme: there were areas where the local clergy encouraged their flock to effectively boycott civil registration, especially of marriages, because they saw it as the government usurping the church's sacrament. We can't really know how many couples simply never showed up at the registry office, but we can see the ones that went through all of the motions until the last step, where they would deny their signatures.)
