The problem is that Blender Guru used Blender version 2.81.12 in the mentioned tutorial, while you use a newer version where as there are some (more then 2 ways to change the amount of displayed wireframe geometry). Mentioned wireframe display, Simplify override as well as display options of modifiers and most likely some i don't even know.
I believe that the easiest way for you should be to activate the Optimal Display option in the subdivision modifier.
I tried version 2.81.12 and 2.82 where the option was off, while in 2.83 the Optimal Display was on by default.
I can't be 100% sure if Andrew Price activated the mode without showing it, but such things are so small and happen often enough off screen.
That said, the Optimal Display should solve your situation.
Also, this option affects each object with the modifier separately.