When they redressed the bridge set and during post-production, they did in fact make other changes to the "Exeter bridge" to distinguish it from the "Enterprise's." First, check out the dedication plaque by the turbolift. While it's not readable, it's noticeably larger than the one on the Enterprise. Second, all the overhead cam lighting is blue, where on the Enterprise it's a mix of colors. Finally, the background sound on the bridge is different (it's the same that was used on the Enterprise bridge in the second pilot as well as for the Constellation's auxiliary control room) than the Enterprise's during the second season. Thus, having already provided several noticeable differences from the Enterprise bridge, there was no reason to also use the high back chair for the Exeter--the contrast was already effectively made. Interesting side point: We know that during its production, Star Trek was perpetually short of money, so they economized at every opportunity. Another great example: while they did produce a unique Exeter assignment patch, they only made them for Tracey and Carter... all of the other 'empty' uniforms on the ship are conveniently placed to avoid showing any assignment patches. Of course, the ultimate example of this for "The Omega Glory" is the creation of a disease that allowed the producers to replace extras with a few bags of rock salt!