I don't really care/know anything about Demons, but I will speak on daemons.
I do not know where this idea that daemons come from the Greek underworld is coming from. Maybe you are confusing modern day ideas about demons and the daemon from Greek religion as I do not really recall any sources suggesting this. I could be wrong, and I would love to hear a source for this.
Daemons in mythology come from Homer and Hesiod. Both of these sources mention daemons. Homer doesn't really make a distinction between gods and daemons. Hesiod however suggests the theory of them being from the Golden Age of man. When the world was destroyed, these golden men stayed as beneficial spirits to man kind.
Socrates was said to have had a daemon follow him and giving him advice about things. The way I read it, it seems he had some sort of clairaudience.
Later on in antiquity, daemons did indeed possess people and cause all sorts of ills. There were even ways in which a person could exspell the daemon.. You can see one of these accounts in the PGM XIII. So, it wasn't "evil Christians" taking all the Gods and demonizing them. Daemons were already slowly becoming associated with things we stereotypically assign to demons. You can dig and find the sources yourself, I suggest "Arcana Mundi" as a good source book.
There are also many other daemons mentioned in the Greek Magical Papyrus (PGM).
Plutarch, On the Ceasing of Oracles, suggests that daemons, not gods, were the ones who gave and maintained the oracles (of course the Oracles still came from the Gods). They functioned as a sort of intermediary between them. Think "Vodou" and you'll understand. He suggests this is for all things in the deity mortal relationship, not just oracles. Since daemons are not really immortal to him, their dying was the reason the great oracles, such as the Pythia at Delphi, were starting to dwindle in use.
There is also the agathodaemon (the noble spirit) which is sort of a guardian angel or a Greek form of a Roman's genius for people in Greek Religion. This daemon protects the household, the oikos. He's given worship on the second day of the lunar month. Usually he's depicted as a snake or man holding a cornucopia.
There's a bunch more about daemons, it's an interesting topic. But, I will leave it at this.
NewEnoch is asking also about Daimons, so I will say that working with one could be found in the PGM, but I would suggest not doing so early in your career. If you wanted to honor the Agathodaemon, I don't think you need to summon it or bind it at all. Just start talking to it, address it, give it an offering (wine, olive oil, bread, fruit), build a relationship. The Agathodaemon is a pretty general term, and in my opinion, it could mean any sort of spirit you see as a personal guardian of the household/you (familiars, guardian angels, etc.)