"Intermediate" is fairly well defined among the classical players I've encountered (many of them teach so this is probably a teachers' consensus), with a little bit of blurring at the edges.

In terms of technique, intermediate means all the building-block skills have been learned but not necessarily mastered:

* comfortable in 3rd position with consistently good intonation, able to play in 2nd, 4th, and 5th positions with OK intonation
* able to play fingered double-stops in tune
* comfortable with vibrato on all fingers
* pleasant tone and deliberate bow distribution
* able to use a variety of bow strokes and articulations: detache, legato, martele, and spiccato

From an ensemble perspective, here's what the Associated Chamber Music Players guide to playing levels says about "intermediate":

"I play with a decent tone. I am comfortable at moderate tempi. I am peripherally aware of other parts. I can sight-read at a moderate tempo but slow down the hard sections."

There's more variation in opinions on what piece in the learning sequence marks entry into intermediate level: some would say the Vivaldi A minor concerto (end of Suzuki Book 4) is the lower end of intermediate, whereas others would say the beginning of intermediate level is as high as Bach A minor (Suzuki Book 7). I'd personally place it at Vivaldi G minor, which is frequently the next piece in the sequence after Vivaldi A minor. Intermediate is a wide range, wide enough that most community orchestra players are considered intermediate. The first "advanced" piece that violinists learn is usually either the Bruch G minor concerto or the Mendelssohn concerto. ("Bruch level" is often shorthand for the minimum standard to be considered "advanced.") When talking about pieces I mean being able to play them in tempo and sound decent; they don't have to be played at professional performance standard but also need to be better than just hacking through all the notes.

I'm sorry that you were made to feel like you didn't deserve a better instrument. That's uncalled for. The way instruments are rated shouldn't imply that at all. A "beginner" instrument is simply one that meets the needs of a beginner, an "intermediate" one meets the needs of an intermediate player, etc. Which is to say, they respond reasonably well to all the techniques learned at those levels. There's no reason you can't start learning on a better instrument. (And many late starters decide to learn the violin when they inherit a violin that's been in the family for a long time, usually not a "beginner" instrument. I did. Nothing wrong with it.)

As for rating ability levels, it's mostly geared toward ensembles: it may take a certain level of ability to keep up with the music they play. Again, that doesn't keep you from joining the group if they'll accept you, but someone below the group's recommended level will likely have a hard time playing the music and may have to fake a lot. (Which I did for years as a beginner playing in intermediate-to-advanced orchestras, because I took a "fake it till you make it" approach. I didn't mind it. But I think it's good to know in advance.)

It can also be helpful in choosing music to buy and work on, as publishers or sellers sometimes list a difficulty level. Many people would prefer not to spend money on sheet music and then discover they are years away from being able to play it.

Of course, if you're not aiming to play in ensembles, then you can simply disregard all categories and just play!

cid said
@Pete_Violin I am thinking more or a school or a group that has limited access or designed for a certain level of ability. Then, I would think the level you are considered to be at by them would be based by their criteria. I don’t know, I was just thinking that might be the case. At one school a wannabe student might be considered not quite their advanced level, but at another music school, maybe they would consider you meet their criteria or ability for advanced? 

Basically, I am just thinking rating as advanced, intermediate, etc might be partly relative by the group or organization you are trying to join. Again, I don’t know, just guessing that level rating might have some relativity in it. Just some thoughts.  

At advanced level, there usually isn't any need to describe it, because everything is by audition at that point. Usually the list of options for audition pieces is enough of a description for candidates to understand what the necessary level is -- you have to be able to play one of those pieces well. (For the most part, getting into any violin performance degree program requires a major concerto, i.e. Beethoven, Brahms, Tchaikovsky, or Sibelius, in the audition.)

If you're not at advanced level, then it's less likely that there will be an audition, and the level description is more for your benefit. If a teacher advertises lessons for intermediate level and up, then you have some idea of what level it takes to be accepted after a trial lesson. Conversely, there are many teachers who teach only beginner level, or only beginner and intermediate levels -- if you're already above the level range they feel comfortable teaching, then you know not to waste your time or the teacher's time. Community orchestras usually only have auditions if their minimum level is upper-intermediate or advanced, so if they say the minimum level is "intermediate" it's to tell you what kind of ability you'll probably need in order to keep up with the music they play. They'll probably accept you even if you don't play at that level.