There seems to be some temptation here on EESE to assume "plates" refer to or equal cells. I believe this is mistaken.
Cutaway images of automotive 12v lead acid batteries I am finding seem to show that each nominal cell consists not of a single pair of plates, but rather of a stack of many interleaved plates, alternately connected to the cell's positive or negative terminal. In effect, each cell is a number of cells in parallel, though sharing the same bath of electrolyte. Six of these cells in series then form the 12v battery.
Likely, the 11 or 13 plate designs are two currently or recently common tradeoffs in the parallelism vs. size or thickness. I'd expect we'd need a battery designer to explain the advantages or disadvantages of each.