When you multiclass, do you only use the highest proficiency bonus of any of the classes you have, do you add together all the proficiency bonuses from any classes you might have levels in or do you use the proficiency bonus that corresponds to your character level from your original class? (For example, a level 13 character starts as a fighter so they use the fighter 13 proficiency bonus.)
Your Proficiency Bonus is based on the total levels you have as a character, not on any one class.
Your proficiency bonus is always based on your total character level, as described in Chapter 1 [p.15], not on your level in a particular class. For example, if you are a fighter 3/rogue 2, you have the proficiency bonus of a 5th-level character, which is +3. (PHB p.163, emphasis mine)
The proficiency bonuses listed in the Ch. 1 table are the same as those listed in each class's advancement table. That is, the bonuses are identical across classes, so there's no differentiation between the "fighter 13 bonus" you mention and the "[anyclass] 13 bonus" or the "[class1] X/[class2] 13-X bonus".