Three different animals, and the distinction is who the lender says owes the money when you're done.
Subject-to: the deed transfers to you, the loan stays in the seller's name, and you make the payments. The lender never approves anything and usually isn't told. The seller stays legally liable on that note. Most mortgages have a due-on-sale clause, so the lender can call the balance due when it learns of the transfer.
A formal assumption: the lender approves you, underwrites you, and substitutes you as the borrower. The seller comes off the note, sometimes with a release of liability. This only happens on loans that permit it, and FHA, VA and USDA loans commonly do while conventional loans generally don't. There's paperwork and a fee, and the terms are whatever the servicer states in writing.
A wrap, also called an all-inclusive deed of trust: you buy on a new note from the seller that wraps around their existing loan, you pay the seller, and the seller keeps paying the bank. The old loan is still in place, so the due-on-sale exposure is still there.
Loose market usage is exactly what your wholesaler did, calling anything where an existing low-rate loan stays in place "assumable." When someone says assumable, ask one question: has the servicer confirmed in writing that this loan is assumable and what the fee is? If they can't answer, it's a subject-to deal wearing a nicer word, and you should price the due-on-sale risk into it and have a refinance or sale plan ready.
