Yes, both factions have both.
DYSTOPIAN CORE
Demon-gods are just the Dystopian Core's version of a higher-up angel. But the main difference is that they do not teach anyone abilities. They simply pass them on passively. Since demons work like a hivemind, the information needed passively gets passed down in the presence of a Demon-god.
Another main difference, is that demon-gods treat normal, lesser intelligent demons as an army, rather than volunteers. So if a demon wouldn't want to participate, they'd be torn apart down to their soul and split to others, more willing to rise up the ranks, to match their god.
The demon-princes are the most intelligent demons out there, which causes them to have the highest amount of respect. Which is why no demon-god alike them challenges them at all, despite everyone being on an equal playing-field. They grant the gift of furthered intelligence, to those they deem worthy to lead their brainless folk with.
Demigods in the demon-world are simply offspring off of powerful or more intelligent-than-usual parents. The bar is set quite low in the Core.
DIVINE SANCTUARY
Gods in the Divine Sanctuary however, are their own folk, not seperated or segregated- but shown respect to simply, with a bow for a greeting, apology or when leaving them, they do not expect anything else, just acknowledgement.
They do band together in groups, to make decisions and ask the general populus to angels, so no group feels excluded.
Demigods are simply offspring or titled as such. If a tutor-angel works hard enough by a teacher's side, they might be crowned a demigod for their hard work and granted advancement in power, as the angel deeming them in question, would see them as responsible enough, not to abuse their newfound powers granted. In the sanctuary, there are less demi- than gods out there.
The full-on gods simply are already born as such in trillion-old families. The angel-gods rarely die, only to demons, is when a member truly perishes. Being the child or cousin of a god is a very big deal in the Sanctuary.