My understanding is that taking out a mortgage later on a fully-owned home (not financed) is accomplished through a cash-out refinance.

I don't know what a "cash-out mortgage" is and how that would differ - care to enlighten?

It's shorthand phrases.
There is actually no such entity as a "refinance". What there is is getting a (new) mortgage and using the proceeds to simultaneously pay off the old mortgage.

Cash-out/cash-back is when the new mortgage is larger than the payoff balance of the old mortgage. The lenders assume (validly) that if the borrower walks out of the closing with cash in his hand, that there is a higher risk.
The lender views a refinance as one lender (the new one) just stepping into the shoes of another lender (the old one). The net risk in the transaction is the same, it's just moved from one lender to the other.

For a cash-back/cash-out mortgage on a house that doesn't have an existing mortgage, the risk is new. There is no "old" lender's risk that gets moved to the new lender. Hence the net risk is is higher.

In recent history, there manifestly was higher risk in mortgages where the borrower walked away with cash in his fist.