It has taken me awhile to get a little bit of an understanding of the history of music and the trombone in the US. Essentially the trombone's "modern" use started in the 1800s probably in tandem with Europe. The US Civil War popularized the cornet bands. These bands consisted mostly of cornets and sax horns. Sax horns were valved instruments which covered all the voices of the brass. Later the cornet bands incorported valve trombones and even later slide trombones. Of course the history of the trombone goes back hundreds of years to the Sackbut but the modern use of the trombone is different since the tenor trombone took over the role of both alto and tenor Sackbuts inclusive. Also the changes in brass instruments with inventions in the 1800s (use of valves and saxhorns) changed the context inwhich the trombone was used.
In my opinion 3 types of trombones other than the bass trombone developed in the 20th century. They were the jazz trombone, the concert trombone and the symphonic trombone. The jazz trombone was a small bore trombone. The small bore gave it a edge which it needed to cut through and or blend with the cornets or trumpets. The concert trombone had a larger bore but not too large and usually included an F attachment. This medium bore horn allowed it to have enough edge for outdoor playing but not as much as the small bore. The small bore section with legit (non-jazz) music can sound like dogs barking which is not a good sound for concerts. The symphonic trombone has a large bore and an F atachment. This trombone had a broader smooth sound which can make an impressive sonorous sound in legit music which makes it ideal for Symphonic orchestras. The F attachment often used on the concert and symphonic trombone allowed these horns to hit several low notes otherwise unreachable.
In summary, The jazz horn cuts through a band but can sometimes sound too brassy. The medium bore concert trombone can cut through without sounding like dogs barkiing and can handle outdoor music well, and the the syphonic trombone plays very smoothly and sonorously and blends well in symphonic orchestras. Student trombones are no differnent than jazz and concert trombones except in quality. The "step up" trombone is just a medium bore trombone with an F attachment built with lesser quality. If these student horns were built with higher quality they would be respectively do the job of a jazz trombone and/or concert trombone.
I omit the bass trombone although it is a fine instrument, because I feel it plays a different role than the standard "tenor" trombone. It has an excessively large bore and large mouthpiece with a different sound like a baritone saxophone in a jazz band or a tuba in a legit band. It is as different from a regular trombone in sound as an euphonium or small tuba.
I see the modern Tenor Trombone comming in 4 types; student, concert, symphonic and jazz. Currently, I have a student horn which I use for practice, a concert trombone, a symphonic trombone, and a jazz trombone. They are respectably the King 606, Ross Reynolds Contempora, Kanstul 970 and a Conn 6H.