Paul,

Understand your intended use-cases - if all you are doing is cutting abstract shapes out of old vinyl records (I did have to google to understand what I think you're doing!) then you can expect a rather more relaxed set of requirements than some of the behemoth machines on this site and build logs. Once you go beyond vinyl onto harder materials then the requirements for the machine build ramp up rapidly - and your experience/satisfaction on an inadequate machine would drop equally rapidly.

For the use-case that you've expressed (and only that!), a cheap Chinese router would likely match your needs, and is a quick way to dip your toes. As others have said they are typically of poor build and specification - the former allows you to understand a bit about machine limitations, and the second shouldn't limit your ability to cut disks. Don't be afraid to consider 2nd-hand particularly if you can rack-up at the seller's door with a blank in hand and demand a demo. Consider your working environment - if indoors near the wife then you'd be clever to avoid the air-cooled spindles (which include the appropriation of wood-working routers)... the noise they make would attract some attention and look closer at the water cooled spindles. Look at the tooling that you'd be using - likely fine carbide cutters which would require a high rotational speed - again, a Chinese water cooled spindle sits well here, but you're not looking at high cutting loads - the entry level 800w spindle would suit.

Much depends on your budget - cutting your cloth, so to speak - from low-end scroll saw to high-end cnc you're talking of costs escalating by several orders of magnitude.

Suggestions to build your own - this can be a good intro to a machine and as your demands are pretty straight forward you could achieve this without any exotic tooling or machinery (get anything that your need machining cut from members here). You could match the machine to your needs - and likely end up with something broadly equivalent to the Chinese routers - although probably better suited to your required working area.