Strictly, a trash-out is the removal of debris and abandoned belongings from a vacated property, priced on volume. Loose usage covers anything that looks like a cleanout, including the one after an eviction, so the two words often land on the same job. The difference that matters to you is legal status of the contents rather than the labor. In a real trash-out the property is empty and the belongings have been dealt with. In an eviction cleanout there may still be a tenant's or borrower's property with notice or storage requirements attached, and those requirements differ by state.
Pricing follows volume, not square footage. Expect a per cubic yard rate, or a per load rate, plus disposal. Deep cleaning after the haul is separate, sometimes quoted per square foot on a vacant house and sometimes as a flat number. So keystone's range for a heavily loaded 1,500 square foot house is the right shape, and the clean afterward is a few hundred more on top.
On who orders it, lenders and servicers usually don't call cleaning companies themselves. They use a property preservation vendor or a field services company that dispatches local crews, takes photos and bills against a set schedule of allowable amounts. Smaller lenders often just keep two local numbers instead, which is cheaper per job and more work per job.
The part to get advice on before you order anything: what a lender may enter, secure or remove while the borrower still holds title is limited by the mortgage or deed of trust and by state law, and it varies a lot. Whether an advance is recoverable is the same kind of question. That's one to put to your attorney on the specific loan documents, not something to reason out from general practice.
