The simple thing is to use the terminology I have used for years:
1. TfB: anything that requires another card to be trashed to give its benefit. This is important because TfB plays extremely differently with no source of +gain. Take something like Moneylender, with +buy I can set up something like Tac/Mlender/Wv and churn a province every turn off drawing 7 stop cards, 3 cantrips and a few dead VP cards. It only costs me $34 worth of setup costs. Likewise with Apprentice, any sort of +gain drastically changes the number and amount of Apprentices I want (e.g. give me two Hoards and I will slap every extra $5 to Apprentice, give me no +gain and I will still get one or two just to thin down).
In contrast, straight trashers play basically the same with or without gains. Chapel is exceedingly rarely going to want additional copies and need +gain for anything. Once I have thinned the (last) trasher becomes a dead card (at least in terms of trashing) and I treat it as a curse.
This also contrasts against straight benefit cards. Perhaps the best contrast here is between Altar and Artisan. Both gain $5s, but the former eventually runs out of dross and more quickly builds up value density of your deck. The latter can bulk out the deck (e.g. Gardens, Silk roads, Keep) on net.
TfB cares about +gains, straight trashers do not. Other benefit cards do not care about (junk) card count, TfB does.
2. Net trashers: your deck ends up having fewer cards than before you trashed usually. This is important for things like Remodel, Rats, Transmute, Upgrade (on Poorhouse boards), and the like which cannot be used against garbage to thin back down normally and are hence excluded. Boards with only non-net trashers play very differently than those with net trashers.
3. Limited trashers: these are the trash cards that can only work with some sort of restriction on what they trash - Mlender, Hermit, Procession. Again, these play differently than unrestricted trashers particularly in response to various attacks.
4. Scaling TfB: These care about the cost of the card. Sometimes the play the same as a regular TfB (e.g. Upgrade vs Altar with the only +gain being Workshop on a potion board), but other times they care a bunch about cost.
As always there are a few edge cases (e.g. Masq), but most often cards play differently depending on the class they are in with Scaling TfB playing very differently from non-scaling TfB a lot of the time. Something that is a net trasher plays differently than a non-net trasher.