If you accept that "the first time through" Cochrane accomplished his first warp flight without the assistance of the Enterprise crew, and without having the majority of his ground crew killed by the Borg, then sure, First Contact definitely represents a change in the timeline.
I think that the reason we didn't see the supposedly extremely important
NX-01 on the Enterprise Dee's observation lounge wall was because originally none of the NX's were named Enterprise. The NX-01 was named Enterprise in the alternate timeline because Deanna told Cochrane the name of the ship in orbit that Cochrane saw through the telescope. Cochrane later request that the "warp five ship" be named Enterprise.
Wouldn't the "first time through" have to have been successful without the Ent-E's assistance? After all, Picard's crew has no record of themselves helping Cochrane and they do have a record of the warp flight being a success.
I like your theory about why the NX-01 isn't on the Ent-D's wall... Maybe the "first time through" equivalent ship was the Yorktown or the Daedalus...
I think the change was a minor one, but it seems to me it is a slightly alternate timeline, unlike J.J.'s movie which is wholly different.