I think wax gets used as a universal term incorrectly. Some of the dispensary providers here might say otherwise but I've always thought that "wax" implied that the product was made with C02 and not butane.
"Budder" is the whipped form of BHO - sometimes it happens as a result of poor purging and sometimes it is intentionally created from properly purged oil.
"Shadder" is the hard, clear, candy-like form of BHO - you don't get clear, clean looking shadder unless you've purged correctly.
Butane as a THC solvent is pretty darn clean and safe when it's done correctly (if you are worried about the health effects of smoking BHO) but I hear a lot of dispensaries sell underpurged oils (I don't live in a med-state so I don't know)