Risks of charging an empty lead acid battery from 10 v
It fell either because it just ran out of juice, or you lost a cell.   The rebounce to 13v after you removed the load is an encouraging sign.   

Charge it slowly and delicately with the smallest charger you have.   Lead-acid and other battery types need a slow tapered recharge when they are in the bottom 20% and top 20%.  

Prepare for the battery to not perform as well as it used to. Lead-acid types do NOT like to be hard bottomed, and the way they take damage is with shortened life and shallower capacity. (Notably, NOT significantly lower voltage, that's a bad cell).  

Lead-acid types are fragile flowers, and are prematurely aged by overcharging, storing partially discharged, bottoming, half bottoming, excessive heat, excessive use, disuse, or looking at them cross-eyed.  If you expect to be abusing this battery regularly, get some nickel-cadmium wet cells (probably a used set no longer fit for use in aircraft, from a FBO)...  abuse rolls off their back and they will outlive you.   I had one set that was abandoned for 27 years, got some vinegar-water and cleaned off all the schmutz, checked their water, and put a charger on them. Performed well for the last 4 years, though they could use an electrolyte change.