The Kitzur Shulchan Aruch writes that in order not to burden the congregation, the custom developed to pray Ma'ariv after Mincha between plag ha-mincha and nightfall, and they were not concerned about the self-contradiction. Perhaps they held that it is possible for the current day to end at nightfall while the next day begins at plag ha-mincha.

Sorry for the late reply.

Exactly that is what happened: I prayed Mincha after plag ha-mincha and then accepted the Sabbath.
After the fact, was that okay?
I made the early welcoming of the Sabbath for the sake of three small children, and otherwise they would not have stayed awake for the Sabbath.

As mentioned above, there are opinions that say yes.

Do you mean that the view permitting it is rooted in the Kitzur Shulchan Aruch mentioned in Y.D.’s reply? I didn’t understand that answer—if possible, could you explain it a bit, please?

There is a tannaitic dispute about the time for Mincha and Ma'ariv. In practice, the Jewish law was not decided conclusively like one of them; rather, one may choose one approach and follow it. What you did matched one view with respect to the Mincha prayer and another view with respect to Ma'ariv. Therefore it is a self-contradiction, and the accepted ruling is that one should not do that. You can pray Ma'ariv early, but in order not to create a self-contradiction, you need to pray Mincha before plag ha-mincha. You prayed Ma'ariv early but did not move Mincha earlier accordingly. But there are opinions—this is what was mentioned in the name of the Kitzur Shulchan Aruch—that since the Jewish law was not definitively decided here, one may choose a different tanna for each prayer separately, even if that creates a self-contradiction. That is what you did.