All are technically correct. The term, "ZIP Code," was originally trademarked, but that trademark expired in the late '90s (source), so the "™" symbol is not necessary (though the U.S. Postal Service still uses the full trade dress for some reason, so I guess they're holding onto their glory days).
Here at MapTechnica, we use the capitalized 'C' (e.g. "ZIP Code,") when referring to U.S. Postal Service ZIP Codes. We use lower-case versions (e.g. "ZIP code," or "zip code") as a generic term when the distinction between the U.S. Census Bureau's ZCTAs and the U.S.Postal Service ZIP Codes is irrelevant.
Just to be 100% clear, MapTechnica does not sell U.S. Postal Service ZIP Code™ boundary data. We sell/use ZIP code boundary data derived from U.S. Census Bureau ZCTAs.
TL;DR: "ZIP Code" = USPS ZIP Code™; "ZIP code" or "zip code" = generic use; ZCTA5 = U.S. Census Bureau ZIP Code Tabulation Area