The RCA Tube Manuals have an excellent summary of tube testers.

Many transconductance tube testers test the tube under static conditions,
using dc on the grid. This is a simpler technique, but not as accurate as
tests made under actual conditions. That is the reason that the best test
is to install the tube in the circuit and see if it works properly. Some of
the very expensive tube testers, like the Western Electric card tester, will
test the tube under conditions likely to be seen in the actual circuit. The
problem with one of those testers is that the tester is one large heavy
suitcase and the cards used in the tester is one large heavy suitcase.

The dynamic tube tester uses ac on the grid and an ac ammeter of the
dynamometer type. Transconductance is equal to the ac plate current divided
by the input-signal voltage.

The summary of the RCA text is as follows:

"The tube tester, therefore, cannot be looked upon as a final authority in
determining whether or not a tube is always satisfactory. Actual operating
test in the equipment in which the tube is to be used will give the best
possible indication of a tube's worth."

Colin K7FM