the only terminology that has much meaning is savage and ultimate.
ultimate is brutally hard and savage is much more difficult than anything else of same range(usually)
there was a time when hard modes were legitimately harder than the standard variant but people complained so much they got nerfed so now "hard" can just be read as higher level. While "expert" is basically just highest level of normal and by level i mean literal item level, not skill level.
And just an interesting factoid imo. Since 'make it harder' was used so many times they don't actual do that. yoshi has said on many occasions that starting with normal and making it more difficult while maintaining the same feel was too big a task so they actually start with savage and take things away to create normal.