I believe that both version of the Enterprise-C are canon. I believe that the Enterprise-C we had seen on Yesterday Enterprise had just been refitted and was going through a trial run to test out it new system when they got the distress call from Narendra III.
Clues that I believe why the Enterprise-C was just refitted.
There was less then 140 survivors on board. Which should be much higher with a ship that carry a crew of 800-900 and haven't receive enough damage for that high of a casualty.
There were no mention of a first officer or second officer.
No mention of a chief medical officer or any medical officers or medics.
Even though there engineering crew and weapons technical crew on board. But no mention of a chief engineer of a chief weapons technical officer.
Only one surviving officer, a Lieutenant (Junior Grade), even though Captain Garrett had survive, but was kill later. There was no mention of any surviving Ensigns. Except for Lieutenant Castillo and Captain Garrett and those who was kill on the bridge. The rest were only crew members.