Rather than use one rule, I'll propose checklist. The more things that you check off, the more likely it's a planet.
- Is it round? Does gravity shape it? (Planets are round)
- If you changed it's orbit to 1 AU, would the planet retain enough mass to stay round? (Things that evaporate easily into gas and disperse aren't planets.)
- Is it below the gravitational threshhold of fusion? (Fusion objects are Stars.)
- Center of gravity is solar body(s).
Possible other indicators
- Does it have an atmosphere?
- Is the orbit fairly circular around the star?
- Is the orbit on the orbital plane?
This doesn't really answer my question: are those outer objects planets? It at least it gives me some criteria that I can apply to the problem, and that is present in my presumptions about what planets are.
Mercury: Missing only an atmosphere. Close enough.
The Moon: No. (Center of gravity is the Earth, and has no Atmosphere)
Gas Giant Moons: No. (Center of gravity is the Gas Giant.)
Pluto: Doubtful Maybe. It has no atmosphere, orbits off the orbital plane, and has an elliptical orbit. Aditionally, I don't know it's composition, but it might not survive at 1 AU.
If we redesignate the Earth/Moon or Gas Giant/Moon as Planetary Systems, then we could have lots and lots of planets. If I move the Center of Gravity to an indicator, rather than a requirement, lots changes.