I bought a li-ion battery off of ebay. The listing claims that the battery has a protection circuit, but a week later, the battery didn't cut out or anything. With a multimeter, it is reading 10V when the battery is listed at 10.8-12.6V. Does this mean the battery doesn't have protection or should the cutoff be lower?
The battery is most likely a 3S Li-ion pack, i.e. 3 cells/packs in series. Protection circuits for single cell Li-ion normally have overdischarge protection set somewhere in the range 2.5V-3.2V per cell, which translates to 7.5V-9.6V for a 3S pack. So this is the range that you should test to ensure that the undervoltage protection correctly triggers.
The listed output voltage 10.8V-12.6V is probably meant to be a conservative estimate of the usable voltage range, since there will be little accessible capacity left below 10.8V (except at very small currents).
Edit To respond to a comment on the question, if you do a search on the model number "DC 12680 protection" you will find other listings (e.g. on Amazon) where it claims to include protection circuitry.
You will also find pages that unmask the typical overinflated capacity claims on these packs. For example, here it is discharged at 1A down to 8.5V and tests at only 25Wh. Notice that this is 3.7V * 6.8Ah = 25Wh, not 11.1 * 6.8Ah = 75Wh. The industry standard is to quote capacity at the cell nominal voltage 3.7V not the pack voltage (here 11.1V). Many unscrupulous sellers exploit this ambiguity to mislead the buyer into thinking that the pack has much higher capacity, here 75Wh vs. 25Wh. Caveat emptor.
Without actually looking in the pack to determine if there is protection circuitry, it would be very difficult to test for the circuitry. The main reason being that in order to do a voltage or current test, you would need to test the extremes( Highest voltage, lowest voltage, max current draw, temp, etc.) and if the battery is not protected you will most likely end up destroying the battery.
Ex: You decide to test if it has under voltage protection, so you start to drain the battery and observe the voltage.
outcome a) The battery has protective circuitry, so as the voltage reaches a low level, around 2.5V per cell(can vary quite a bit, determined by the manufacturer), the output shuts off and you suddenly read 0V.
outcome b) There is no protective circuitry, so the battery keeps draining and the voltage reaches a range where you can determine that there is no protective circuitry and the voltage is still high enough that the cells have not completely died, around 2V-2.5V, depends on chemistry.
outcome c) There is no protective circuitry, and the voltage on the cells becomes low enough that they start plating lithium and become useless.
It then becomes a game of trying to achieve outcome b while avoiding outcome c. If testing is the only option to determine if it has protective circuitry, I would recommend draining the battery to about 2.4V per cell or so, and if it gets that low with out shutting off then there is no circuitry, at least none of any value. You need to be very careful not to go much lower, because all it takes is one cell dropping below 2V to make the entire battery useless. I would not recommend test over voltage or current, as these can easily lead to the battery "rapidly disassembling".
TLDR; It can be done by very carefully lowering the voltage, however it should only be done as a last resort as cell damage can easily occur if not careful.
re: "...battery has a protection circuit..." What kind of protection? Over charge protection (max current &/or above the cutoff V), over discharge protection (max current &/or below the cutoff V), max current protection or all three? If you want to know for certain which type/s of protections your battery provides, your battery will need to be monitored (V & A) while charging & discharging it.
If the li-ion cells are 3.7 Vnom, then the max charge should be 4.2 vdc per cell an 12.6 vdc for 3 cells that are connected in series. If the charging current cuts-off before the in circuit charging V exceeds 12.6V, then your battery has over charge protection. If the discharge current cuts-off at or before the battery reaches typical cut-off voltage levels (~3v per cell), then your battery has over discharge protection.