I don't think it's really clear that
Tessa started out as a short form of
Theresa. I can't find any evidence that
Theresa or
Teresa was in use outside of the Iberian Peninsula before St.
Teresa de
Avila became famous in the 1500s. But
Tessa was in use in Italy a couple of centuries before that. The Italian name dictionary I own,
Santi e Fanti by
Enzo La
Stella T., says that
Tessa was regularly used in Tuscany in medieval times, and that it may have indeed been a short form of Contessa, because Conte, the Italian word for "count", was definitely used as a male given name in Italy during the same medieval time period.
Tessa became fairly common in Tuscany partly because of
Monna Tessa, who founded the Francisan Oblates in
Florence. She died in 1327, and so predates
Teresa of
Avila by two centuries. La
Stella says that in modern times
Tessa is used in Italy as a shortened form of
Teresa, but that that is a
new usage, not based on the medieval use of
Tessa as a name in Tuscany.
http://www.oblatedimonnatessa.org/italia/inglese.htm
And the form
Tessa seems to have been introduced into England in the 19th century by novelists who used it as a name for Italian characters.
So all in all I think that
Tessa is probably a name with two separate origins, and it is just as correct to say it is a medieval Italian name, probably a short form of Contessa, as it is to say it is a pet form of
Theresa.