False. There were two versions: XE and XE-G.
All Corollas had AC, except for the fact that XE had it without economy mode whereas XE-G had two buttons instead of one, don't know how it worked back then but one read "Eco" and one read 'AC'.
The one with black bumpers was XE, it had two spoke steering wheel and simple dashboard. XE-G was the one with colored bumpers, it had three spoke urethane steering wheel and wooden garnish around the AC vents on the dashboard.
Then came GL version which had additional stuff like central locking and power windows as standard, plus a better stereo system with four speakers and a tachometer inside the cluster.
GL was the same as 1.6 GLi except for driver side airbag, four spoke steering and 4AFE Efi under the hood.
You missed the XE-G trim.