All but the most basic iMac's come with a dedicated graphics card rather than an integrated Intel Graphics card. For day to day use (browsing the internet, word processing) that is more than sufficient. But as you have clearly indicated in your first post, you want this computer to edit 4K video. With that requirement in mind, you are going to need substantially more computing resources than just a beefy CPU. Editing 4K requires a lot of resources on the graphics card side of things, which is why almost all of the suggestions in this thread are get a good graphics card. It also generally requires faster and larger storage arrays as 4K files are large.
As it stands, your budget just does not make sense for the requirements you have outlined. If you buy a Mac Mini, baseline price is $800. That said to make it work for editing 4K you are going to need to purchase an external graphics card ~$600-$1200 and given the extremely limited internal storage on the base Mac Mini you will need to purchase additional storage (say at least ~$200). Going by that a Mac Mini configuration suitable for 4K editing is already hitting ~$1600-$2200. This isn't to say you can't build out a system for less, but the configurations you have suggested don't make sense.
