You write; "So if my level 5 Paladin's effective cleric level is 5 - 3 = 2", but in your own quote it clearly says; "A paladin uses her level as her effective cleric level when channeling positive energy."
So what is it that you think is wrong?
Did you, by chance, look at the Spells entry?
Yes, your caster level for your spells equals your paladin level -3.
However, your effective level for channeling equals your full paladin level.
There's no 'turns undead like a cleric who is x levels lower' derived clause present in Pathfinder.
Oh wow. For some reason I could have sworn Paladin Channel was level minus 3. I've skimmed the ability multiple times while looking at Paladin stuff. I could have sworn I read it. And now I looked again just now, incredulous, and I've been wrong all this time... maybe I got that from looking at Hospitaler and then just generalized the -3 part of it.
Oh wow. For some reason I could have sworn Paladin Channel was level minus 3. I've skimmed the ability multiple times while looking at Paladin stuff. I could have sworn I read it. And now I looked again just now, incredulous, and I've been wrong all this time... maybe I got that from looking at Hospitaler and then just generalized the -3 part of it.
A lot of classes have the option to gain channel energy but typically gain it at a reduced level, as you expected. In fact, because Paladins generally have more reasons to increase charisma, they are very often a stronger channeler then clerics (although usually with a reduced pool of available channels). The Sacred Servant can even use their Divine Bond to pump the channeling power even further, putting their individual channels well beyond what almost any typical cleric build would have.
It's an unappreciated feature of the Paladin, it's often more economical to use the two uses of Lay on Hands to recover after a fight or to spend the round to recover everyone's health after a large AOE (especially the favored tactic of all DMs of lighting off a fireball during an ambush).