Axminster or Proxxon. Small professional lathe or toy. Just my opinion.
Judging by the description in your other thread I think you are wise to get rid of your current lathe which seems like it is beyond economic repair.
The lathes you have suggested (especially the Axminster) are great if you only ever want to make pens and other small spindle work. For not much more than the price of the Axminster precision pro you could get a Record Power DML320 VS which would give you much more swing over the bed, the option to fit a scroll chuck and the ability to make much bigger projects. Unless you think you will never make anything but pens or are extremely restricted fior space this would be a better choice in my opinion. I have reviewed both lathes for Woodturning mag. Both are good, the Record is more versatile.