An air carrier crew taxiing to EWR Terminal A on Taxiway A was told by ATC to hold before Taxiway RB and then told by Ramp to continue to their gate via Taxiway A; but ATC objected stating ATC controlled movement on A.

Narrative

We landed on Runway 4R. We exited at K and were told by Tower to turn left on P and hold short of Runway 4L at E. After two departures on 4L; we were instructed to cross 4L and contact Ground; which we did. We were finally cleared by Ground Control to 'taxi to the ramp via Alpha.' As we proceeded down Alpha; the First Officer left COMM 1 and established communication with Ramp Control on COMM 2 as I monitored COMM 1 on ground. As we rounded the corner on Alpha and approached RA; Ramp advised my First Officer to hold short of RA due to another carrier aircraft which was about to taxi from the ramp via RA but had not begun to move. My First Officer switched back to Ground and advised them of Ramp Control's instructions while I switched to Ramp to maintain communication with them while monitoring Ground. Ground then confirmed Ramps request to hold short of RA; which we did. A minute or so later; the other carrier aircraft still had not moved. Ramp then advised us to 'proceed to the ramp via Alpha and advise Ground;' which I acknowledged; and my First Officer advised Ground as we began our taxi. As we started our movement as instructed; Ground Control questioned why we were moving and reiterated that Alpha is under their control even though they ceded that control twice during our taxi. We proceeded down Alpha to the ramp and to the gate as instructed by Ground and Ramp. I contacted the Tower from the jetway and tried to straighten out the confusion. My impression was that Ground and Ramp have had discussions about this previously and he mentioned that he would call them again. Several factors were involved that contributed to miscommunication. First; having the First Officer switch to COMM 2; call Ramp; advise them of our location; get instructions; then come back to COMM 1; and tell Ground what Ramp wants; does not seem like a good idea while the Captain taxi's in an extremely congested environment; like Newark. Second; Ground and Ramp should communicate between themselves and not rely on Crews to be the go-between making the Ground to Ramp transition more seamless. Third; there are no taxiway/ramp boundary markings or spots that would simplify the transition process. For instance; a quick look at the PHL 10-9 depicts clearly defined boundaries; spots; and frequencies; reducing ambiguous instruction and confusion.