Where does the Maximum Emitter-Base Voltage come from?

"Emitter Base Voltage" is the maximum voltage that may be applied when the base-emitter diode is in reverse; not conducting. This is generally much lower than a small signal diode in reverse can handle.

You suspicion is correct - it's emitter-base, which means it's the (positive) voltage from emitter to base, not base to emitter (which would be given as -5V)
So basically it means you can't let the base drop >5V below the emitter (or emitter 5V above the base ;-) )

Your guess about the order of the terminals is right: it is the maximum reverse voltage that the emitter-base diode can block. Some people claim this makes a good zener diode , others use the zender mode as noise source.