Solution 1:

In the US, these phrases have different meanings. Pay out would not be part of a purchace on installment; pay out is what a company does to distribute funds.

Payment - the individual amounts paid toward the total owed.

Payoff- the final payment, or the amount that if paid now would be the full amount owed. (Payoff can be one word as a noun in this jargon, but as a verb it is “pay off”. “What is my payoff amount? I want to pay off my loan and be done with it.”

Pay out is unrelated.

Here is a link for payoff: “Payoff The amount that will pay off a loan in full.” https://www.wellsfargo.com/mortgage/tools/glossary/

Solution 2:

Your feeling on these terms seems broadly similar to mine. A pay out having the potential to be ongoing; perhaps dividends on shares or perhaps some sort of bonus payment. The only occasions I've heard the phrase 'pay off' it has been used to indicate the end of the line somehow; perhaps a redundancy payment, or buying your way out of a contract or other obligation.