Sizing Ground wire for 200AFeeder to House
No, you're interpreting the Code correctly
Table 250.122 in the NEC is what governs equipment grounding conductors, and for a feeder with a 200A maximum breaker ahead of it, it requires a 6AWG copper wire, which is what you are installing.  This is because the grounding conductor only has to carry current long enough for the breaker to trip; in normal operation, there should be 0 current on it, as opposed to the hots and neutral, which have to sing for their supper, 24/7/365.
Note that this is a different table from the table that's used to size grounding electrode conductors such as those going off to your ground rods.  Your 2AWG Cu GEC is a trifle undersized for 600 kcmil Al service conductors according to Table 250.66, which'd require 1/0AWG Cu for that application; however, since you're dealing only with ground rods and not a water pipe or miscellaneous electrode, NEC 250.66(A) kicks in and says that a 6AWG conductor is all that's needed.  (A concrete-encased electrode would fall under 250.66(B) which only requires a 4AWG conductor, and ground rings require no more than a 2AWG as NEC 250.66(C) requires the GEC to be no larger than the ring conductor, which is required to be 2AWG minimum by 250.52(A)(4).)