The term is ambiguous, and many uses don't include the trailing dot. See IETF RFC 8499 section 2.
> DNS is strictly ASCII-only
DNS is not strictly ascii. At the protocol level labels are octet strings. You may be thinking of the LDH convention.
• Hmm, I wasn’t aware of that admission of ambiguity. Allowing that fuzziness in terminology puzzles me, because if you don’t have the trailing dot, then what you have fundamentally isn’t fully qualified. Sure, some resolvers may treat it as though it were, but some won’t, and are perfectly correct not to. (I haven’t the foggiest idea what the balance of implementation behaviours might be. I know enough DNS to be dangerous, but I don’t live and breathe the stuff.)
• I should have said DNS hostnames are ASCII-only, which I believe to be true. (And yeah, this still depends on conventional rather than rigorously defined terminology.)
As far as client-side tools... no one ever uses the trailing dot and that can lead to some interesting situations if you have a search path set and use a resolver that resolves wildcard. Use of the search path at all is fairly unusual for "client-ish" setups though. You could also view the common non-use of the trailing dot as one of the causes of the ICANN recommendation against top-level domains resolving, as the search-vs-hostname-vs-fqdn ambiguity would be less common (but still present) if people commonly used the trailing dot.