Your assumption is incorrect. So, the variances in your stock thermo and your 3rd party thermo will vary with both:
A) the amount of thermal mass in the smoker itself. An empty electric smoker's stock thermo will read "off" by a different number of degrees compared to a quality thermometer when empty than when packed full of meat. Therefor you must test and take note of the average variances when the smoker is mostly empty, average amount of full, as well as packed. This is you getting familiar with your smoker.
B) all smokers have "zones". If you setup 6-8 thermometers in your smoker, you will find that you have both hot and cold zones based on their location in relation to the burner and exhaust. You will want to familiarize yourself with these areas and plan accordingly. Perhaps by rotating foods periodically throughout the smoking session or by some other means.
Tips:
1. Fill your water pan with sand or gravel or something else with density and wrap the whole thing in foil. This will help to even out temp swings and zones within your electric smoker.
2. Research other modifications that other members with Bradley electric smoker's have made to even out temps. Many members have your type smoker and your type frustrations and have found workarounds.
3. Have you ever looked on a chart to see how electric ovens actually hold a temp? Remember, an electric smoker is just an electric oven you put smoke in. An electric oven simply holds an "average" temp based on how it is set. If you set an oven for 300, it's probably actually running up to about 310 or so, then turning the heat off until it hits let's say 290, then running back to 310. Rinse, lather, repeat. Those numbers are just examples.
So if you're setting your smoker to 225, you're most likely spending a good deal of time below that temp which is at minimum dragging out the time of your smokes. So when you're making sometbing like 3-2-1 ribs which are based on time and not temp, you will run into problems.
The point being that setting an electric smoker to 225 degrees, in my opinion, isn't necessary.
Always cook to temp and tenderness. Let time be the only unknown variable. It'll be done when it's done. Good words to live by and the main tenant of bullet point number 4!
4. Have fun, drink beer, and don't overthink it. It's not science, it's just science ;-)
These are all things that will become first nature to you as you practice, practice, practice. Today's failures are tomorrows chili's if nothing else.