It seems the dedication plaque wasn't present in "The Cage" and first shows up in "Where No Man Has Gone Before," so we can perhaps assume a new plaque is given before every five year mission (sure, why not?) or at least updated at the whims of the Captain (a la the Titan novels which implied that the Captain chooses the quote).
The Black and Silver Dedication Plaque was either April's or Pike chose it for his first five-year mission, and the updated bronze one I guess we can assume he chose for his second five-year mission. Maybe related to an update of the bridge (he likes those) and to match with a lighter color scheme. The main difference is that it specifies what type of Starship Class the ship is (
), which some ships do (Shenzhou, Discovery), and some ships don't (Franklin, Enterprise in various timelines). Kirk, upon taking over or before his five-year mission, will keep the gold/bronze color scheme, but remove all the pesky names and go for a bare-bones aesthetic, which seems to become the norm until the 2290s at least (Enterprise-B brought back names, although it was a lot less and more organized than the long lists seen in the NX Enterprise and Discovery era ships).
I once went into a long theory on the origin of the "where no one has gone before" quote (that I can't find), presuming at the time that it comes from Kirk's log entry at the end of Star Trek VI which was affected by the events of that movie and the lesson of having a less-than-human-centric attitude going forward. Unfortunately, the Enterprise dedication plaque from Discovery Season 2 disproves that notion, and it seems that Pike may have either coined it, or someone else did between 2119 (when Zefram Cochrane seems to invent the phrase "where no man has gone before") and ~2256. Perhaps Kirk is quoting Pike in his log entry in Star Trek VI, or at the very least, the dedication plaque at the time he took over the ship before having the quote sandblasted off (and eventually replaced with the correct Cochrane version of the quote). Maybe Pike had the quote changed to "where no one..." when he realized he needed to get used to having more than just men on the bridge.
The Black and Silver Dedication Plaque was either April's or Pike chose it for his first five-year mission, and the updated bronze one I guess we can assume he chose for his second five-year mission. Maybe related to an update of the bridge (he likes those) and to match with a lighter color scheme. The main difference is that it specifies what type of Starship Class the ship is (

I once went into a long theory on the origin of the "where no one has gone before" quote (that I can't find), presuming at the time that it comes from Kirk's log entry at the end of Star Trek VI which was affected by the events of that movie and the lesson of having a less-than-human-centric attitude going forward. Unfortunately, the Enterprise dedication plaque from Discovery Season 2 disproves that notion, and it seems that Pike may have either coined it, or someone else did between 2119 (when Zefram Cochrane seems to invent the phrase "where no man has gone before") and ~2256. Perhaps Kirk is quoting Pike in his log entry in Star Trek VI, or at the very least, the dedication plaque at the time he took over the ship before having the quote sandblasted off (and eventually replaced with the correct Cochrane version of the quote). Maybe Pike had the quote changed to "where no one..." when he realized he needed to get used to having more than just men on the bridge.