Did ancient people understand divinity in the same way we do today?

"Ancient people" are a very broad group culturally, geographically, and temporally. Their beliefs were extremely varied and rich, and probably included most things people think today, especially since many of our ideas are inherited. Here are some general strains, ordered from least to most personal / individualized:

• No god. Despite our modern idea that the ancient mind was superstitious, there certainly were atheists. The Bible makes reference to this (Psalm 14:1), but it was a current idea in Greece as well, according to one of Plato's dialogues (Gorgias) in which Socrates repeatedly suggests that his interlocutors will consider Homer's stories about gods to be fairy tales or old wives' tales, but he doesn't.

• A universal presence or panentheistic god. The belief that the world is suffused by, or included in, some divine or metaphysical nature. The divinity in this case is not personal. Characteristic of Buddhism and Daoism. This is also closely related to the idea of mystery religion or mysticism, becoming aware of deeper reality, and/or moral law attained by the wise.

• It might be fair to link this to animism, as in some North American Indigenous peoples, where the natural world (earth, plants, animals, phenomena) have souls — or you might fit that in at the bottom of this list, depending on how you see those souls as participating or separate from a divine omnipresence.

• A deistic creator god. Responsible for the origin of the world but not an active force now. In another Platonic dialogue, Timaeus, the eponymous character talks about the god behind the existence of the universe. This kind of god is associated less with immediate phenomena and more with reasoning about a prime mover and the sorts of characteristics he might have that are reflected in creation. This also introduces the idea of a heavenly (or at least pre-universe) existence.

• A theistic god. Like deism, but the god is still active in the world. Nevertheless, the god still holds some special place above material law. It makes sense to worship such a god because he sometimes responds. Can also be the head god of a pantheon. Dwells in some sort of heaven, whether that idea grows out of the sky itself or is abstracted away from physical space. An example might be the Egyptian Ra, the sun and creator.

• An assembly or pantheon of gods. These gods have jurisdictions over certain realms or events. They multiply as a function of their activities... for example, how they turn the tide of war or how they influence natural phenomena. (This doesn't necessarily mean they are solely responsible for how the phenomena behave, but they can supersede the normal operation of the world.) They are active in the world and have personal characteristics and dramatic stories. They are pleased or displeased by human action, e.g. sacrifice or disrespect. They can also be associated with planets or stars, which were also thought to have influence. They can have factions. Examples: the Roman, Greek, Egyptian, Hindu, and Norse pantheons.

• Territorial or national gods. These are associated with a people or place, or even a token or totem. They are stronger or weaker, and they protect and fight for the tribe that serves them. They can have particular needs that people must serve to appease them or keep them strong. In Mesopotamian literature, for example, humans are created to alleviate the gods' drudgery.

• Deification. Some humans were considered deities, particularly in function of their kingship, either alive (Egypt) or after death (Rome). Emperor Vespasian when he was dying is said to have remarked, "I think I'm becoming a god."

Interestingly, some cultures had no problem with mixing and matching. Ancient Greek religion is famous for this. Plato talks like a deist when discussing the nature of the universe but also invokes the Olympian pantheon headed by Zeus, themselves subordinate to Fate. Before the Olympians were the Titans, who were the children of the primordial gods that first emerged from Chaos. Meanwhile, demigods and mythical creatures populate the world and interact with humans, often amorously. It's a real potpourri.

Since your question makes reference to conceptions of the Judeo-Christian God, an aside on him. We also see a huge variety of forms, or at least a development.

• There's certainly a creator god (deism). Within a couple of chapters he's actively making choices about humanity (theism) and "walking in the garden" with them.

• He later meets Abraham and Moses in some physical form as well.

• He institutes moral law and is "the judge of all the earth".

• He can act as a territorial god that consistently overcomes other territorial gods (as in Exodus against deified Pharaoh or when the Dagon statue falls over before the Ark of the Covenant). This could also be a way of saying that they're not gods at all, as Isaiah writes and as in Elijah's contest on Mount Carmel.

• There's some polytheism, at least of servants of God; in Kings there's an assembly of spirits of whom one is sent to give Ahab some bad advice, in the psalms there's a reference to the assembly of the gods, and in Daniel an angel is detained "fighting the prince of the Persian kingdom" until Michael, "one of the chief princes", rescues him. There is of course also Satan, said to be a former angel.

• In the extra-biblical tradition, including Kabbalah, and equally in the New Testament, panentheism and mysticism are evident: "God is love", "God is him in whom we live and move and have our being". (Incidentally, Kabbalah also has animistic elements such as rock golems.)

• Meanwhile, mainstream Christianity has certainly deified Jesus. Of course, this is the source of long, bitter, and ongoing schisms: Is God one, as in the Shma' Yisra'el confession, or three persons, as in the Trinity?

It would be fair, then, to say that this God includes all the categories above, with the twist that unlike Greek polytheism, all of the categories are said to be same eternal entity or subservient to him. (Iain Provan of Regent College points out one difference between Genesis and the Babylonian creation myths it responds to, namely that God never has to contend with other divine beings.)

There are paradoxes wrapped up in that — some people consider it scandalous to say that the omnipresence "in whom we live" was also a particular human being — and whether you judge this to be the result of a patchwork tradition by many editors or simply the manifold revelation of something ineffable is a matter of doctrine not suited to this forum.