Both of those lasers use the M2 Nano controller, which LightBurn does not support. Try to find one with a Ruida, Trocen, or TopWisdom controller, or you’ll have to swap the controller for something like a Mini Gerbil or Cohesion 3D.
Preenex and Orion Motor Tech both sell decent K50’s with a Ruida in them, but the price point is higher - they’re going to be in the $1300 to $1500 range. The $340 K40 link you showed is decent enough, but you’d need to swap the controller - A C3D LaserBoard would be my choice, as the setup is quick and quite painless, and it’ll allow you to expand to using a rotary and a Z table if you want to add those later.
The C3D board is $200 USD is about 166 Euros (according to Google). You don’t need anything else, just the board - it’s plug and play. Having said that, if you are not in the US, you’ll pay shipping and import taxes and fees that will increase that price. The Mini Gerbil board is less expensive to start with ($90 USD) and they have a European distributor, so the import fees won’t hit you. You would buy that from Awesome Tech. Both C3D and Awesome Tech have installation guides and instructions to show you how to do the installation.
Be aware that at this price point, you have scraped the bottom off the bottom of the barrel, so the machine is best described as a kit that has been assembled for you, but you’ll still need to adjust, tinker, and fix. Just making sure that you have appropriate expectations - these machines do work, and can work very well, but quite a few corners have been cut to get the prices this low.