Actually, no, most of them do not list a specific trainer as a prerequisite for entrance to the class. Some certainly do, and allowing entrance into those without that trainer would, naturally, be a houserule (and often a good one, in my opinion). But you can be an Archmage as long as you have the right skills, feats, and spells. The vast majority of prestige classes do not have a prerequisite indicating you must be inducted by a trainer, so requiring one is a houserule.
I'd also say that having prestige classes from multiple books does not necessarily indicate that yours is a cheesy build. I ran an absolutely terrible build a while ago that used a homebrew Vampire template, Races of Stone, the Expanded Psionics Handbook, Tome of Battle, and PHB. Why? It let me build a 600 pound vampire whose primary form of attack was waddling forward and then suddenly leaping through the air, only to bounce off his target back into the space from which he initially jumped. I called him Mr. Bombardini, after a /tg/ thread that gave me the idea. Anyway, point is, he was, by a wide margin, the least effective character in the session - at one point, he wound up gnawing on a pipe organ, although admittedly this was because of a natural 1. I think I hit once? Ever?
Anyway, as I was saying before I got distracted, I needed all those books to actually construct a character that felt like Mr. Bombardini. It was a silly game, yes, but this is just the first example I found in my list of recent characters. I needed to be unusually large, and the best way to make that happen turned out to be a Goliath Psychic Warrior, and I needed a lot of combat feats to make the charging at all viable (I only hit once, but by Jove, when I did, Dracula felt it), so I needed a Fighter dip, too. And in order to make it possible for him to inexplicably throw himself and have the returning property, I needed Bloodstorm Blade from ToB, so there was some Warblade to get the prerequisites.
In fact, I'd say that if you have a specific concept (as the OP does), lots of multiclassing can be very helpful in meeting up with your ideal. You might be able to do something with a modified Geometer, for instance, although it's an arcane casting class for Wizards in Complete Arcane. In order to make that work with a divine caster, you'll either need a permissive DM or a lot of shenanigans that are bleeding edge to make an Archivist's spells count as arcane spells (you'd probably use an Archivist, from Heroes of Horror, because they have a spellbook). If you want your Runepriest to have a bit of melee capability, you might want to look into picking up a few levels of Crusader, from Tome of Battle, so that you can smite your foes with the hammer of your god. See if your DM will let you retrain maneuvers whenever you increase your initiator level, in order to make things simpler (it sounds more complicated, but it means you don't have to worry about when you take the level).
To the OP:
A lot of that probably is more complicated than you're looking for. It's probably actually bad advice, because I'm throwing around all sorts of terminology and books than you're able to find and quickly process. So, don't feel like you actually need to look all that stuff up - it's meant to serve as an illustration that you can use a variety of sources to help sculpt a basic concept into the specific one you really wanted to play. You're just getting into this edition of the game, so don't worry too much about complexity - you can run a Dwarven Cleric from 1-20, and it will be a solid character that can contribute at all levels of the game. If it also covers everything you need your character to do, that's perfect. Remember, don't be afraid to write your own flavor descriptions of your spells - your cure light wounds might take the shape of a glowing sigil traced on the brow of your ally, helping you keep that runic connection even if the spell doesn't specifically mention it. If the selection in the Player's Handbook isn't doing it for you, look into the Spell Compendium, which should provide enough extra goodies to keep you in good shape for several games. You probably won't require much more than that, if you even need one splatbook.
Protip, though. If you play a Cleric, remember that you don't necessarily have to be the party healer first and foremost. I don't know how 4E's climate is, but a lot of 3.5 first-timers assume that the Cleric is only there to keep the party's hp topped up - remember, you can spontaneously convert spells into cure spells, so you don't need to prepare them. You can fill those slots with spells that improve your (or other players') ability to beat face, if you want, or reserve a few for better-than-standard healing spells once they become available.
Yeah, Rune domain on a cleric is probably a solid, elegant way to go. Good luck, buddy.