No, you left out (1) which says:
(1) It retransmits only communications from GMRS stations operating under authority of the individual license under which it operates; and,
Because it says “and”, both conditions must be true. The only example of (1) that I interpret would be if I set up a repeater and only my family members operating under my call sign use the repeater. (2) means that they still must properly ID using my call sign while using the repeater. Both must be true. If the repeater is ONLY used for that purpose then there’s no need for the repeater to (separately) transmit my call sign.
But in my opinion I believe that as soon as someone using a different call sign uses the repeater, then the repeater is required to ID itself periodically either using voice or CW.