Well, I had a revision B motherboard (something I specifically requested, because of certain features I wanted to experiment with), and the extended 80 column adapter, which expanded my system memory to 128k (bank switched, since only 64k was addressable), but the CPU in my system was definitely not a 65c02.
My system also did not have the MouseText characters that came out with the //c, so by the link you are referring to above, I had an unenhanced Apple //e. Nonethless, both the logo on the case and the startup logo said //e, not ][e.
This website refers to a model that was discontinued in 1985, and is right beside an image that looks exactly like the model that I had. Note that it has the //e logo on the case cover. It's entirely possible that it was called the ][e for a very short time after launch, but I had never seen it... and I was practically living in a computer store near my place at the time, when I was preparing to get my own system.
It's all pedantry anyway. But sounds like you had what could be called a "partially enhanced" machine:
If you are able to turn the machine on, the easiest way to identify an
enhanced IIe is to look at the machine name printed on the top line of
the startup screen:
Apple ][ indicates an unenhanced IIe
Apple //e indicates an enhanced IIe
The catch is that you might have a machine which has been partially
enhanced: it is possible for the CPU, video ROM and firmware ROMs (CD
and EF) to be updated independently (the firmware ROMs must be a
matching pair). Looking at the chips would be safest bet.
I remember engaging in many online (BBS & Usenet) discussions where the common shorthand was to use ][e for unenhanced, //e for enhanced (for times when it mattered.)