Does it mean abandonment of
a religion altogether, or just the rejection of the existence of God?
"abandonment of a religion" assumes that the person was affiliated with some particular religion to begin with, and he or she has abandoned it.
"abandonment of religion" means that the person has given up on the concept or idea of religion, perhaps after serious examination of an array of possibilities. Not everyone is born into a religion. An atheist family would not bring up their children in any particular religious faith. But when the children are able to make decisions of faith for themselves, they might decide to join a religion, or abandon the entire concept of religion.
But technically atheism comes from 2 Greek roots: "a" - meaning without, and "theos" meaning "god". So literally, it is defined as the belief that there is no higher power. It is part of the word family with monotheism (mono: one), and pantheism (pan: many). Then there is the Latin-derived word "deism" - from "deos" meaning "god". The similarity of theos and deos is not accidental! Deists believe that there is a higher universal power, but they do not believe according to the teachings of a formal religion.