Truly, I do not know what this thread is about.
Grounded systems / circuits are referenced to earth. There is zero current flow to earth because of this, except for very rare and anomalous events.
Connected would imply a closed circuit and current flow with the earth, but the earth path is connected at a single point to the grounded (referenced to ground) system, so the earth is connected but the path through the earth is an "open circuit".
In order of strictness of NEC requirements:
Grounding electrode conductor: A wire from point A to point B with no splices or spliced using listed irreversible splices. Must be "accessible". Point A is the busbar where the system bonding jumper is located, point B is the "nearest" listed grounding electrode.
Equipment ground: A wire or raceway listed in 250.118, required to carry fault current and facilitate OCPD tripping. Also is connected to earth by convention, is bonded back to the source, which is bonded back to the GEC and the earth..
Bonding: Required to be able to carry fault current, but the path can be through the accessory metal structures, metal enclosures, strut, and other methods not listed in 250.118.
If the requirement says "ground it", the methods are stricter (250.118 materials). If the requirement says "bond it" the methods are looser, metal other than 250.118 methods may be used.