The following is adapted from this Agora post by Tezos Co-founder Arthur Breitman. As noted in the post, there is no official body which can authoritatively set Tezos nomenclature, but the following is recommended:
Used somewhat interchangeably, either as a noun or an adjective to designate:
A open-source project and software (as in, “contributing to the Tezos codebase”)
A peer to peer network of nodes maintaining a blockchain (as in “a Tezos node”)
The specific Tezos chain with the most economic relevance (as in “the Tezos chain”). Typically, this would be today the chain whose millionth block had hash BKtC4QCWoF73kxLj773vFpQuuwrnye6PS7T1aM3XEPvFXiQbNu7.
To be avoided:
Tezos is not a unit of cryptocurrency. “I sent you 1 Tezos” is incorrect
Tezos is not an entity. As a decentralized network, Tezos can not partner with a company or institution.
A unit of the cryptocurrency native to a Tezos chain (e.g. “I sent you 2 tez”). Tez is invariable. It is not capitalized (except at the beginning of a sentence or when you would otherwise capitalize a noun). “I sent you 2 tez” and not “2 Tez”.
“XTZ” is an ISO-4217-compatible code for representing tez on the most economically relevant Tezos chain. Unless there is a very specific reason to use an ISO code for it, the term tez should be preferred. Situations where the ISO code might be useful typically involve accounting systems, exchange rates with other currencies, and anything that might need some sort of standardized code. Generally speaking, as of the day of this post, “XTZ” is vastly overused in situations where “tez” or even “Tezos” would be appropriate.