What Do These Non-Glossary Terms Mean?

The "difference" is in the intent of the person doing it. If grandma polishes her coins, then puts them in a yellow paper envelope for a decade "to keep them safe", is she deliberately trying to "retone" the coins? No. Then it's regarded as "natural". But if I did exactly the same thing, it would be "artificial", because I know that yellow paper retones cleaned coins, so my goal in doing this would be to try to retone the coin.

This is the problem with trying to use the "natural" versus "artificial" distinction, since the difference is entirely with the intent of the person, and "intent of the person" is not recorded on the coin itself. The chemistry is identical, and the results are identical.

A more useful definition is "natural" versus "accelerated", because "accelerated toning" uses chemicals that a coin is extremely unlikely to encounter by accident to produce that decade's worth of toning in mere seconds, minutes or hours. Bleach, sodium sulfide, sulfur jelly (Deller's darkener) and other noxious agents all cause "toning" that might look realistic, but is chemically distinct from "natural" toning, and will leave telltale residues on the coin that can be detected if you've got the right equipment. The other problem with these accelerated methods is, like Frankenstein's monster, they can get out of control, go unstable and wreak havoc. An accelerated-toned coin might look good today, fresh out of the coin doctor's box, but in a year's time it could very well be a corroded, ugly mess.

The term Lapped Die I am familiar with through Overton's Early Half Dollar Die Varieties. It means over polished die. It is noticed when some devices are removed, as in the "single leaf" varieties of the Capped Bust half dollar series.