I have a mini-lathe.
When I talk about my mini lathe you should be aware that I'm talking about one of the 7" X 10"/12"/14"/16" models.
I've spent the last year working on my 7x14. It was a Weiss WBL1835 imported by Amadeal.
Weiss is one of the three main Chinese manufacturers that make mini lathes. The other two are Seig and Real Bull.
Precision Matthews (who don't sell mini lathes or mini mills but do sell larger Chinese lathes and mills) used to use Weiss as one of their main suppliers; I don't know who they use now.
So, my mini lathe wasn't a bad example of the type by any means.
I've learned so much during this last year fettling, adjusting and upgrading the mini lathe. I've had to learn a lot of bench work (if you don't have a mill, and even if you do, a certain amount of the work that is necessary to bring out the potential as a useable machine tool in a mini-lathe requires bench work).
I guess I don't regret the knowledge I've gained and I have really enjoyed the process but a few things are important to know:
The 7" swing Chinese mini lathes, out of the box, are seriously lacking in important areas that make a lathe usable for anything apart from trivial work in plastic or aluminium (and even aluminium can be tricky on these lathes, out of the box). They lack rigidity, they lack precise quality control and fitting, they lack a decent work envelope. The absolute best examples will still require some fitting and adjustment and the majority will require a fair bit more work. The worst are little more than scrap, often sold by importers who are out to make a fast buck.
The money and time you will spend sorting out the above problems out of the box will, at regular points in the process, make you (quite reasonably and sensibly) regret starting out.
There isn't a single mini-lathe you can buy that, even fettled, adjusted and upgraded to within an inch of its life, will have the capability of something like a tidy and well looked after Southbend 9 or other smaller 'old American iron' lathe.
If you absolutely need a really small benchtop lathe due to space and it has to be new, the 8" x 16" swing Chinese lathes might be worth a look. Their one big advantage is the width of the bed, which tends to be 4" rather than the usual 3". That adds a fair bit more rigidity into the whole system.
LMS don't seem to do an 8" x 16" lathe and the nearest alternatives LMS sell (the 8.5" x 20") are significantly more than you seem to have set for your budget.
At this point, you are left with a choice between:
Buy a 7" x 14" Chinese mini lathe, cross your fingers that the importer you chose will have a good relationship with the factory and doesn't just buy the QC rejects and will provide you with good customer support. Then spend a fair chunk of time and money on making it into a useful, pleasant to use, machine tool, that you may find works great for you and you enjoy using for years...or slightly less ideally, ends up being too small for what you end up wanting to do with it as you progress...or in the worst (but least likely, being honest) case, never becomes any kind of useful machine tool and is abandoned as a waste of money.
Or
Save a fair bit more money and go up a size in swing to 8" or more. That will tend to get you a better Chinese lathe. It will still need some time spent adjusting it for best performance but hopefully you won't be actually doing the fit and finish that should have been done at the factory.
Or
Be patient, watch a lot of YouTube videos about the fundamentals of a lathe, browse a lot of the topics on here and keep an eye out for a tidy, well-cared for used US/European lathe on FBM, eBay or Craigslist. The advantage of this is that it likely will work out cheaper overall as it is likely to come with tooling and you will get a lathe that will keep pace with you for a fair old time on your machinist journey.
The latter really is the ideal but it does require a bit of patience and you (or someone you know) to be able to reasonably assess a used lathe's 'health'.
Assuming you don't mind your first machine tool being a project in and of itself and you have the patience and money and a bit of a natural feel for machine fettling and fitting, then the first choice is okay; for some people the challenge makes it worth it.
The middle choice is probably the 'safest' choice but probably the most expensive.
