Naming
In this section
What is it?
A conical-bore B♭ brass instrument in the tenor–baritone range — and the tangle of names (baritone, tenor tuba, saxhorn) that surround it.
Why do Americans call it a baritone
American school-band tradition absorbed the euphonium under the looser label 'baritone,' partly from earlier smaller-bore instruments and partly from the manufacturing and pedagogy of US school bands. The instrument is usually a euphonium regardless of the name on the case.
Is a euphonium a saxhorn
Historically yes — the euphonium descends from the B♭ saxhorn family patented by Adolphe Sax in the 1840s. Whether today's wide-bore euphonium still counts as a 'saxhorn' is a matter of how strictly you draw the family line.
What is a tenor tuba
'Tenor tuba' is what orchestral scores call the euphonium. When Holst, Strauss, or Wagner wrote for tenor tuba, a euphonium is what plays it — read in concert-pitch bass or tenor clef.
Euphonium vs baritone — what's the difference
In British usage they are two different instruments: the euphonium has a wider conical bore and darker, fuller tone; the baritone horn is narrower and brighter. In American usage 'baritone' is often just a loose name for a euphonium.