There is a distinction between pseudorandom generators, as defined in theoretical computer science, and pseudorandom number generators in general. Whether it's a useful distinction to make for tags on this site is a matter I will, at least for now, leave for others to argue.
(Unless somebody chimes in with "yes!", I'm somewhat inclined to assume "no" by default. But I'm also perfectly fine with leaving them separate, if someone finds the separation useful.)
As for the other tags mentioned above, I do agree that prg should certainly be a synonym for pseudo-random-generator, just like rng and prng already are for random-number-generator. I also agree that on this site csprng should be a synonym of random-number-generator; this is Crypto.SE, so any on-topic discussion of (pseudo)random number generators pretty much has to concern those that have, or are claimed to have, at least some level of cryptographic security.
(There might be an argument to be made for having a separate tag for "true" hardware RNGs, as opposed to PRNGs, but so far nobody seems to have created one. I'd say wait and see; if there really is a need, someone will create it.)
I also agree that randomness should remain separate, since it's mainly being used for cryptographic schemes that consume random numbers instead of generating them. Of course, there will naturally be some overlap in questions between this tag and random-number-generator, but that's perfectly fine and to be expected.