What is the difference, really? (In a recreational context, not technical diving.)
. . .
It seems others have cleared the OP's confusion on this, but I'll take a stab anyway.
The terms "primary light" and "secondary light" (also, and perhaps more commonly, referred to as a "backup light") originate in technical diving, and were never widely used in recreational diving. If they have any meaning in the context of recreational diving, then it's the meaning imported from technical diving: Your primary light is the one you generally use, and if the primary light fails during a dive, you switch to a backup light. For a technical diver, being able to see inside a cave or a wreck is generally critical to a diver's survival. However, it may not be quite as critical for a recreational diver, since the recreational diver can always just call the dive and surface if he has insufficient light. Since light is not quite so critical, it is common for a recreational diver to carry only a single light and not carry a backup light. Of course, carrying a backup light on a recreational dive is still a good idea.
The light manufacturers and dive shops that market lights for technical divers will use these terms with different lights. The ones marketed as suitable as primary lights for technical divers are robust, powerful, and have long burn times. The ones marketed as backup or secondary lights are more compact and will have shorter burn times and are typically less powerful, although with improvements in technology this is not always so. As far as technical specifications, the line between lights marketed as primary and lights marketed as backup is getting fuzzier every day.
Your confusion may stem from the fact that recreational divers often choose to buy lights they intend to use as primaries but which were originally marketed to technical divers as backup lights, since they are often powerful enough and have long enough burn times for recreational use.
For my recreational diving, I carry two identical lights. I guess the one in my hand would be considered the primary, and the one clipped to my BC is my backup.