This seems a really simple way to remember which is which, but if it was right, I'd expect an equally base name for the one for Naturals than semiring ("rig" doesn't count). I think this is how it goes for operators:
semiring: $+ *$
e.g. Naturals, no inverses
ring: $+ - *$
e.g. Integers, semiring with additive inverse (enabling subtraction)
field: $+ - * /$
e.g. Rationals, ring with multiplicative inverse (enabling division)
Are there any wrinkles I've missed that mean this isn't quite right?