I live in a house that was built in '62. I am having to put in new water service from meter to house and in basement where old galvanized line comes in there is a ground from the breaker box attached 8 feet from the floor and about 20 feet from the breaker box. I found another ground in the flower box outside the house several years ago when planting azaleas. This is directly behind where the breaker box is located inside the house. My question is... can I safely remove the ground on the water line and put it through the wall to another grounding rod in the same flower bed? The electric meter is on the other end of the house and should have its own ground. Right?

No, you cannot. You must re-attach this bond to the house's copper water piping.
Right now, the wire serves as both a water bond and grounding electrode. If you replace the galv pipe with plastic you lose the electrode (underground water pipe) but you must keep the water bond.