They are clearly not "light speed" as we saw from "Broken Bow" (We could see the carrier beam hit first before the actual energy beam rode on top of it in that temporal chamber room)
That wasn't a "carrier beam." Remember, everything that happened in that room was echoed/doubled. You'd see a person start to move, and then there'd be a second image that would be delayed by a beat before it started to move. You'd hear echoes when a person talked. And, by the same token, you'd see a phase pistol beam get fired twice.
King Daniel Into Darkness wrote:
The phaser and photon(ic) torpedo effects in Enteprise were exactly the same as the ones in TNG, DS9 and VOY, weren't they? I recall thinking it odd in-universe that Starfleet would change the look of their weapons a couple of times in the 23rd century (the Abrams' movies' effects are based on the ones seen in TMP and WoK), then switch back to the way they used to be - and incredibly lazy IRL that they ignored the way they looked in TOS and the classic movies and just reused the old effects for a show set 200 years before.
There you're confusing the way the filmmakers chose/were able to represent the effect with how it would've looked in-universe. It stands to reason that the less sophisticated FX animation techniques of the '60s would've produced different results. Roddenberry himself saw the designs and FX of TOS as merely the best approximation he was able to achieve with the available techniques, which is why he was happy to change them when better techniques came along (as in TMP, where he redesigned the Klingons and asked fans to accept that they'd always looked that way).