That's the etymology of "incel", but in practice it's not just a definition, it's a community identity with a lot of associated baggage. There are plenty of lonely young men who don't consider themselves "incels", often because they don't want to associate themselves with that community.

And if you examine the community, you'll quickly find that they don't think sex work "counts". Because what they want isn't sex, it's the status they imagine it confers, and if you had to pay for it, you didn't "earn" that status.