"Spirit", although now typically used in a generic and ostensibly culturally neutral way, had four major senses in ancient thought:
(1) In pagan philosophy, Greek pneuma or Latin spiritus was thought to be an especially fine substance, which played an important role in the functioning of the body and in the workings of steam-powered machines. In Stoicism, pneuma was thought to be not only the medium of the brain's control over the body, but also the stuff of the so-called World God, the active part of the universe. In late Platonism, it was also thought to be the material of the so-called vehicle of the soul, connecting it to the body. The idea was that the finest material was closest to the immaterial.
(2) In Greek-language Judaism, pneuma served as the translation of Hebrew rûaḥ, based on their common everyday meaning of "breath, wind". Unlike in Stoicism, there was no idea that the pneuma of individuals was divine in the sense of being parts that in concert make up God. Rather, pneuma comes from God, but is sometimes thought of as a part of living creatures (as with Adam), sometimes as an additional influence from God, in other instances again as an intermediary entity (or activity) through which God inspires people.
(3) In Christianity, a novel sense emerges - but one that draws on both the pagan and the Jewish sense -, according to which the immaterial is spirit, and so God, angels, demons and human souls are wholly spiritual or partially spirit, partially material.
(4) In late antiquity, pneuma was also used - e.g. in the Greek Magical Papyri - as a term for individual beings, also called gods or daemons, but without much terminological precision. How much this comes out of Judaism, how much out of pagan thought, cannot (I think) be known with any certainty, because of the strong influence of Jewish thought on pagan magic. But it is important that, as I said, spirits did not emerge as a kind of entity with their own profile, nor did pneuma become a popular umbrella or a clear synonym of daemon. It also did not imply incorporeality.
Unfortunately, while it is useful to know these different meanings to historicize our own usage of the word, it does not help us very much with the later history of the term, because the neutral sense of spirit - which needn't be either an angel or a demon, but might be - comes from (just) outside the Greco-Roman world. The appearance of ritual magic invoking spirits or spiritualities (rūḥāniyya, related to the Hebrew rûaḥ), often associated with the elements or the stars rather than with God, angels, or demons, was a novelty in Arabic-language literature produced in Iraq around the 9th century, and while the connections to earlier Greco-Roman and local Mesopotamian practices (Jewish and non-Jewish) are evident, their exact nature is obscure.
Both in Islam and - after a large number of translations from Arabic to Latin introduced the idea to Latin Christendom around the 12th century - in Catholicism, the existence of spirits distinct from good angels and evil demons (or devils, or jinn) was a matter of controversy. Some, in a defensive tone, argued that they were angels, others thought they were deceptive devils.
In conclusion: spirit is not a culturally neutral term or a close equivalent of the Greek daemon.
(And incidentally, although daemons could be contrasted with gods, the general term for both was god [θεός], so it is rather disingenuous to divorce pagan philosophy about daemons from polytheism and associate it with a trend toward monotheism. Similarly, the switch from, say, describing the Japanese kami as false gods to calling them spirits is barely an improvement.)