Q: Should I locate my thermometer in the center of the cooker? -or- why do my two thermometers read differently?
No not necessarily. On a UDS or vertical cooker, you have to decide if you want to measure above or below the cooking grate. (in the food area or below). Those areas can vary greatly. On a horizontal offset, do you want an average temp or a reading on the left and on the right. Remember, the hottest area will be closes to the coals, unless you use a baffle.
So, it can vary depending on where the coals are, if you have a baffle, a water pan, minion method or not, where the smoke stack is, how much meat is on there, if you set it above or below grate. There are many factors, what type of fuel you are burning, is fuel evenly distributed, and outside air temp to name a few!
Bottom line, a thermo itself is accurate. The internal smoke temps can vary greatly within the cooker, depending on all of those factors. Furthermore, it can vary from one thermo to the next.
A thermo is a good way to tell if you are steady and near your desired cooking temp. It also usually indicates a good average temp, but smokers rarely stay balanced across the whole chamber. you can add a LavaLock Products tuning baffle plate or a FireDial to help attain better balance, efficiency and control.
Additionally, A BBQ Vortex will deflect heat into certain area (in)direct or direct, which can unbalance temps, when it makes sense to do that, like in the case of air frying crispy wings or direct sear. https://vortexbbq.com/
On a horizontal offset cooker, you can put a thermo on each side. On vertical cookers, one above the grate and one below is helpful as well, or one at the upper grate and one at the lower. Then you can tune your cooker with dampers and baffle systems. This is why two thermometers can read differently.
Check out some GREAT adjustable thermos here.
LAVALOCK® BLACK FACE BBQ GRILL THERMOMETER
and baffles here
BAFFLE PLATES (HEAT DEFLECTOR TUNING)