When somebody requests a beam-out, they say:
[officer’s name] to [ship], [amount of people] to beam up
I know the AI would know to route this channel to a working transporter room after the word “beam”, but why not follow this format instead?
[officer’s name] to [transporter room], [amount of people] to beam up
Do you mean why don’t they specify which transporter room they want? Like they should say, for example:
Commander Riker to transporter room three, one to beam up
I’d suggest that crew on a planet don’t necessarily know which transporter room to specify, or indeed where the on-ship crew will decide to beam them to.
Officers do, I think, specify that they want to be beamed to sickbay when it’s a medical emergency. But in general it’s probably up to the ship which transporter room to use, or indeed whether to do something else entirely.
(Contradicting my entire answer, I think sometimes in the show they do specify a transporter room. I’ll check later on, or someone else will and will deservedly get the rep points.)
Few reason, which changes from TOS to TNG.
The first is that TOS had a switchboard operator setup. In the Enterprise, all calls come into Lt. Uhura who processes and redirects as needed. Intra ship communications are also like room to room intercom systems. This reflects on the technology familiar to the writers.
In TNG, it gets more automated. Instead of room to room, it's person to person. The computer does routing, there is no dedicated Comm person.
The second is likely the most important. If the away team calls for the transporter room, which transporter room? Which room on what ship? What if there is more than one ship in the region? By calling the specific ship with a beam out request, the context and destination is already known. And in TNG, the "x to y" syntax is needed by the comm badge computer to initiate the communication channel, much like "Computer" is the syntax keyword to initiate a comm badge to ship computer information request. (Siri or Ok Google in real life).
third, No need to know the low level details of which room or who's on duty. Maybe the room they think is free is full, busy, or down for maintenance. Are shields up? Red alert? What if the transporter room is empty?
Finally, the request to the ship is done because the command staff also needs to be aware, especially during dangerous away missions. While some of the more mundane scientific research or shore leave transport request can just be scheduled or directly requested from the active transporter room, Away missions are too random and important for this.
