^ Nailed it. A major factor in determining a ship's class is its capabilities, not just how much she looks like another ship.
From where I sit, the refit we saw in TMP made the Enterprise something of a prototype, changing her from a Constitution class ship to a one-of-a-kind vessel, i.e., Enterprise class. After seeing how well, or how badly, the various changes worked out in the field, designs were rethought, resulting in more refined refits on other old Constitution class ships (almost certainly including the USS Constitution herself, since the class name is retained, and, of course, the ship that would become the Enterprise-A).
In other words, the TMP refit was the buggy prototype, the E-A was the regular production version, hopefully free of the prototype's oddities.
From where I sit, the refit we saw in TMP made the Enterprise something of a prototype, changing her from a Constitution class ship to a one-of-a-kind vessel, i.e., Enterprise class. After seeing how well, or how badly, the various changes worked out in the field, designs were rethought, resulting in more refined refits on other old Constitution class ships (almost certainly including the USS Constitution herself, since the class name is retained, and, of course, the ship that would become the Enterprise-A).
In other words, the TMP refit was the buggy prototype, the E-A was the regular production version, hopefully free of the prototype's oddities.