Practically speaking, I would suggest to think of 'intermediate' as starting with a player who can tackle the easier pieces that you might sometimes find major players toying with - Lagrima might be common example, and that category would carry on up to but short of the kind of pieces such major players start having to work a bit, so most of the easier Bach (e.g. Allemande to BWV 996); from this point I would suggest the aspiring player, assuming they can tackle this repertoire with confidence, accuracy, good sound and sense of style, becomes 'advanced'. Being able to play the very major items of repertoire to a very high standard and learn them quickly at that, one would probably be 'professional', quite possibly 'virtuoso'.
Simply being able to play the pieces of an intermediate level through probably isn't quite enough, one would wish for the same qualities as mentioned for the 'advanced'.
And the word I would use for one who is pre-intermediate is 'elementary', I only use 'beginner' for one who has literally recently started. Unless they are making extremely stately progress, after a year or less I would no longer use 'beginner'.
If want to look up grade exam lists such as ABRSM, I would have grades 1-3 as Elementary, 4-7 as Intermediate, 8 - diplomas as Advanced, post-diplomas (e.g. FRSM, FTCL) Professional, whether working as such or not.
In the end it doesn't matter, every small step is an achievement in itself, and the satisfaction for ourselves and anything we can communicate by playing for others is reward enough, compared to arbitrary designations of level, none of which are water-tight.
Simon Ambridge Series 40 (2005)
Trevor Semple Series 88 (1992)
Louis Panormo (1838)
Alexander Batov Baroque Guitar (2013)