Australian here who bought a car in a similar situation. Here's how i did it for the Australian redditors...
Initially, the advert said it had finance owing but we took a look anyway as it was an economic late model second hand for a reasonable price.
Girlfriend really liked the car and given the k's, i was also particularly happy. So we did a revs check (now called a PPSR - or a personal property securities register check). We do this anyway - regardless of how honest the owner is. Chances are their honesty is detracting from another issue (rebirthed, recent stat write off, etc). A risk in no way worth taking.
The certificate noted nothing against the vin/plate other than financier's trading name and certain details. This allowed us to contact them with general queries such as payout details, loan break costs, paying the balance as a private buyer etc. Get all details! Names, time/date if you called, or keep copies if you emailed. This makes life easier if someone down the track decides to be difficult.
After the queries (The financier can't give much info to a third party), we asked the seller for the most recent statement details and how much was owing.
We payed the balance to the finance company's account - not the sellers, and made sure to get the payment receipt numbers (the invoice would've been mailed to the seller - do not wait for paperwork; take as many details as possible of your actions).
Finally, we transferred the registration over to our names and we left.
Unfortunately we couldn't avoid the uncertainty of waiting for the finance company's acknowledgement - but thankfully it was fairly quick. They sent it to the seller whom i hassled for. Once i received it, all was done.
A primary reason i was confident in all of this was the sellers character - she was moving cities which checked out and was generally very honest with us - but it was not an opportunity to be complacent and never should be. Normally, if there were any strange signals or lack of transparency, know when to walk away. They haven't stopped making cars yet, and there's always a seller.
If you have trouble with emotion, spend a few weekends at car yards, normalise the fact you are buying a car, and logical, rational decisions start to come naturally.
Hope it helps