Then, what is a NUMA processor?
It's a processor capable of acting as a NUMA node
is it a processor where cores are grouped, with each group of cores sharing some level of cache?
No, not really, it's literally just any kind of CPU capable of acting as a NUMA node.
To me, NUMA is a concept which is appropriate for clusters.
Again, sorry but nope, NUMA is almost always a single-server-architecture. Basically just about any 2+ socket server you can buy now fits within the NUMA scheme. It's why we server geeks spend so much time ensuring we lay out our memory and IO devices in a balanced way and make sure our OS's are configured to know they're working in a NUMA system too. Anything else means your CPUs spend way too much of their time just acting as a memory or IO controller.
Of interest have a look at HP's 'The Machine' and IBM's 'Watson' architectures - they're still very much in their infancy but I think are the way we'll be doing things in 7-10 years.