I suspect there's a confusion about what a province is during the Republic here.

Provincia is the Latin. The first definition of that word in the OLD is:

the special function or task assigned to a magistrate, originally one holding imperium.

Imperium is the power of a magistrate to give orders and see them carried out, largely associated with military command. A commander laid down imperium when he returned to the city. It's largely a military authority.

The vinc root seems clearly connected to vinco, to conquer. I don't have a Latin etymological dictionary on hand, however. Whatever the case about the etymology, the fact is that governors in provinces during the Republic were always on campaign. That was the point; a province was a military occupation of a foreign land; even the supposed pacifist Cicero (a completely false accusation, by the way), conducted campaigns and expected to win a triumph for his military action in Cilicia.

In Caesar's case, he conducted the administrative business of Cisalpine Gaul during the winter, leaving him free to campaign during the proper season, and on at least one occasion had to rush back to Transalpine Gaul to manage a crisis. This is not too different from the normal practice of the Roman consuls during the bulk of the Republic; usually they'd settle domestic affairs in the first few months of their term, and then spend the summer months on campaign. While Sulla's reforms put far more emphasis on the consuls' legislative duties (and hence keeping them in the city), the proconsuls (the usual title for a Roman governor) continued to operate as generals first and administrators second, if at all (Pina Polo's book The Consul at Rome is the best treatment of the pre-Sullan consulship I know, if you want a reference).

Augustus reformed this system of provinces, largely stripping senators of military command, and placing his own legates in provinces that required military occupation. Those senators, who used to be generals, took up more administrative matters, as you can see if you look at Pliny's letters to Trajan.