A few points:
The "grounds" you are referring to are properly called Grounding Electrode Conductors, or GECs. They are a bit different than the typical ground run with a circuit, the Equipment Grounding Conductor, or EGC.
If your copper water line is buried outside for at least 10', then it is a grounding electrode, and you need to connect a GEC to it within 5' of it entering the building. If the buried metallic portion is less than 10', then it is not a grounding electrode, and you instead run a bonding jumper to the metallic water piping system, anywhere along its length. In either case you jumper across any short non-metallic breaks, e.g. dielectric fittings at a water heater.
1 ground rod alone, or 1 ground rod and a water pipe electrode, is not sufficient for the NEC. You need to add a 2nd ground rod, or another electrode like a Ufer (concrete encased electrode, foundation ground).
At the service the GECs terminate to the grounded (neutral) service conductor. This can be anywhere from the weatherhead (for an aerial service) to the main service disconnect, or in between. Connecting them at the meter is at discretion of the utility, and practice varies by region; some utilities prohibit it, some expect it.
One GEC has to be unspliced, e.g. a continuous copper conductor from the grounding electrode to the grounded service conductor. If you have one unspliced GEC, then you can have splices in any of your other GECs. That means if your unspliced GEC is as large as your other GEC or bonding jumper, you can just bring the latter to the unspliced GEC and connect them together. Note that the required size for a GEC to a water pipe electrode (or I think for a bonding jumper for a water piping system) depends on the service size and may be larger than the minimum allowable size for the GEC to a ground rod (which need not exceed #6 Cu)
So if your utility permits it, and your water pipe is a grounding electrode, you can bring both GECs to your meter. You can also just bring your water pipe GEC or bonding jumper to anywhere on the the GEC to your ground rod, as long as the latter is as large as as the water pipe connection.
Cheers, Wayne