We regularly leave a deposit on a horse, and have received them too. It's normally 10% of agreed purchase price, subject to a satisfactory vetting. If the horse/pony fails the vet, then you get the deposit back, otherwise you are expected to complete the sale. if for some reason, you decide not to proceed having put the deposit down, then you forfeit the money.
If the pony is advertised as ovno, then they are expecting an offer on the asking price. I would offer lower than you are prepared to pay, and then that give you room to raise the offer should they turn you down! EG on an asking price of £1000, offer £800, meet at £900 
We normally pay with cash (and get paid with cash) on collection. Usually, collection is within a few days of the vetting - you might get charged livery if you take too long to collect, but you can discuss that with the vendor.
If they agree to a lwvtb, then get a written agreement drawn up, that both parties are happy with, stating length of trial, details of insurance, what type of work/trial the horse/pony is to have, all contact details, and how/when payment is to be made, among other things.