5 mW and above, but less than 500 mW, is Class IIIb. But the rules sound about right.
Unless something changed in recent years, lasers 1 mW or more but less than 5 mW are Class IIIb unless the beam characteristics are such that it is not reasonably foreseeable for 1 mW or more to enter the eye and get focused onto a single point on the retina, in which case the laser is Class IIIa. However, I see many laser pointers in the 1-5 mW range labelled as Class IIIa.
Class I visible lasers have power of less than .4 microwatt.
I decided after posting this that I oughtta check to see what 21 CFR
1040.10 says nowadays...
That regulation has tables with all sorts of exposure limits for the various laser classes.
As it turns out, visible lasers from 1 mW on up but less than 5 mW, according to the limit based on output power is Class IIIa regardless of power density in the beam.
However, there remains a regulatory distinction (in the USA) between Class IIIa lasers with 2.5 or more mW/cm^2 "irradiance" (beam power density) and ones with less. I believe the test conditions include a 7 mm circular aperture to simulate a fairly dilated pupil of a human eye - which means that a Class IIIa laser that cannot get .9621 mW or more through a 7 mm diameter round hole does not require as harsh a warning label as one that can if I read the regulation correctly. Ones with irradiance less than 2.5 mW/cm^2 must be labelled, "CAUTION - Laser Radiation - Do Not Stare Into Beam Or View Directly With Optical Instruments". I have yet to see such a laser. Class IIIa lasers with irradiance 2.5-plus mW/cm^2 must be labelled, "DANGER - Laser Radiation - Avoid Direct Eye Exposure". This includes all laser pointers 1-4.99 mW that I have ever seen.
I do remember back when 2 mW HeNe lasers sold by Edmund Scientific were mentioned as Class IIIb. Looks like those are IIIa now.
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