Not all non-profits are corporations–many non-profits are trusts, and legally speaking trusts are not corporations (although it is not uncommon for them to own corporations, or to have corporations as their trustees). Many small non-profits are unincorporated associations. Nonprofit versus for-profit status is an orthogonal dimension from corporate vs non-corporate legal structure.
Also, "non-profit" isn't a "colloquialism", it is an official legal term in several jurisdictions: for example, 37 US states have (at least partly) adopted the American Bar Association's "Model Nonprofit Corporation Act" [0]. And it isn't just an official legal term in the US, see section 48 of the Australian state of Victoria's Payroll Tax Act 2007, entitled "Non-profit organisations" [1]
[0] https://en.wikipedia.org/wiki/Model_Nonprofit_Corporation_Ac...
[1] https://austlii.edu.au/cgi-bin/viewdoc/au/legis/vic/consol_a...