*Technical geek-hat mode activated*
To the best of my knowledge, it's Time Lord biology, not the TARDIS, that determines the quota of lives. The TARDIS provides an environment conducive to a smoother regenerative transition, but isn't strictly necessary for it to happen, nor is the TARDIS responsible for "counting" how many lives have regenerated.
River handing the Doctor her regenerations was a ploy to counteract the poison she'd given the Doctor, which had been stated to be so fatal that NO regeneration was possible, so presumably all her regeneration energy was used to remove the poison with no change to the Doctor's own regeneration count.
For that matter, if you want to count regeneration energies being used, the Tenth Doctor regenerated without changing...funneling all the energy into his spare hand in Journey's End so that he remained the same, but it could arguably "count" as using one up in terms of what the Doctor is biologically capable of. I would almost count Eleven's aborted regeneration at Lake Silencio, except that turned out to be a fireworks display by the dumb chameleon/robot/ship thing (although for some reason some fans do count that). The Tenth Doctor claimed to have "given up" some years of his life to recharge the TARDIS when it was stranded in the Cybus-world, and the Eleventh Doctor explicitly used some regenerative energy to fix River's broken wrist in the Manhattan Angels episode.
Of course, I ultimately end on the same point that they do...the show isn't going to get hung up on the Doctor having a limited set of lives; as long as there's any intention to continue the series they'll find a way to get around it and keep it going. Heck, I'm fine with taking Eleven's offhand remark in Sarah Jane's show about being able to change 500+ times as truth if it means we can ignore the limit. And if I really wanted to get out-there in speculation-land, the last episode had the Doctor somehow "entering his own timestream" (as if it's a place you could get to), and presumably like Clara experiencing his earlier selves...it could easily be conceptualized as a move similar to the Valeyard's in Trial of a Time Lord, where somehow interacting with his own past (even paradoxically) could somehow grant the Doctor a renewed set of regenerations "taken" somehow from his previous incarnations. Though it doesn't even have to be so convoluted...in The Five Doctors, the Time Lords try to bribe the Master into helping them and the Doctor with an offer of a new cycle of regenerations, and we've even seen that they somehow did just that when bringing him into the Time War (before he escaped to the end of the universe). Which means there must be some Time Lord technique or technology or something that could presumably reboot the cycle, so all that need be done is handwave a way for the Doctor to experience it and move on with a fresh 12 lives available (at the end of which it's done again, and so on).
Honestly, there are plenty of "outs" to go with this "issue", and I don't think anyone in the fandom really expects it to actually be a problem if/when it comes around.