One thing that I like about the more generic dieties is that it opens up the possibility of having a single diety feel different in different cultures.
Think of it as being along the lines of how whenever the Romans conquered someone they would look at their pantheon and say stuff like "Hey you know your Wodan, he's a lot like our Jupiter. Thor... hrm... seems a lot like Vulcan, or maybe Mars to us. So whevever we put up a temple to Jupiter you can just assume that it's also a temple to Wodan."
This means that the various gods could be worshiped across cultures, just keep the names the same for simplicities sake. If you feel like making up entire pantheons for every different culture that works too but really a lot of the concepts of the gods in the PHB and DMG are pretty cross cultural.
For example, among a tribe of barbarians Kord might be the father of the gods, but that doesn't mean he's the only one worshiped. Maybe they worship Meloria as Kords wife, the Raven Queen is the goddess of winter, Pelor the god of summer, Gruumsh is the god of war, Veccna is the god of magic and witchcraft, Moradin is Kords smith, ect.
By contrast a civilized country might worship Pelor as the leader of the gods, Kord as the god of war, ect ect..