On screen evidence the refit Enterprise was a heavy cruiser: http://movies.trekcore.com/gallery/displayimage.php?pid=40646&fullsize=1 Enterprise schematic from Star Trek III when McCoy entered Spock's quarters.
And from "The Starfleet Technical Manual" no less. Said source was also used in TMP, as the USS Entente is also canon, and Star Trek II & III made the Saladin, Hermes, Ptolemy and Federation class canon. Although not much of the rest of it is entered into the canon in any form, and the NCC registries are refuted by TOS-R, which is why I would warn against using ship names and numbers or the number of Connies in service.
Actually, with that in mind, we do have a wider perspective on the canon fleet of the TOS era. The USS Reliant was also at Starbase 11, meaning the Miranda class existed in the TOS era as well. And the USS Merrimac (NCC-1715) is canon, and though the canon does not deal with what class it is, we could infer it is a Constitution class ship.
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