Second-hand, I wouldn't worry too much about the type of lathe. When something is advertised look it up on lathes.co.uk for detailed discussion of various models and makers.
BUT! Condition matters much much more than the type of lathe when buying a pre-loved machine. Bear in mind that old machines may have been thrashed, abused, dropped, left in a damp cellar, and have bits missing. Spares may be difficult to find, expensive or unobtainable. No matter how good a lathe's reputation, I would never buy a second-hand machine sight unseen. Ideally you need to check it for wear, damage, and see it cutting metal, and know what to look for. Cosmetics can be very misleading – a new paint job may be bad news! It may take some time to find a good example of what you want at the price you are prepared to pay. Much easier if you are the kind of chap who enjoys doing up old machines, and have good contacts.
A particular downside of Myford Lathes is they attract premium prices, sometimes ridiculous. Other makes don't generate the same excitement, probably only because they are less well known.
Over the last 30 years large numbers of bigger better manual machine tools have been dumped on the market because industry and education have moved to CNC. Many of them are in good condition and much cheaper than when new. The downside is they may be too big for a home-workshop, require 3-phase (not difficult), and might be in poor condition. You can derisk this by buying from a dealer, but don't be surprised they need to make a profit.
A new western lathe will be out of your price bracket. Very few hobbyists cough up for them.
Last option is the extensive range of Far Eastern lathes available new. For many people this is the low risk option. You have consumer protection if you get a dud, delivery is organised for you, you can select machines based on size and need (eg imperial or metric), short delivery time (usually), and you get a lot for your money. The downside is these are hobby machines not finished to the highest standards. However, this is the route I went, and – for what I do – the machines I got are definitely 'value for money'. All of them worked out of the box, all benefited from minor fettling, and I've had a lot of fun from them.
Sadly, it's not possible to steer newcomers to the Whizzbang Model B Toolroom Lathe of 1955, assembled by caring master craftsmen, which does everything, lasts forever, only costs £50 delivered in perfect nick with a full set of accessories, and will sell for £5000 next year! Instead, you have to decide what you want and navigate the risks.
Dave