The chalumeau register, whether in the clarinet where it's standard terminology or in the saxophone where it's rarely used, simply means the register of a woodwind where the pitch is the fundamental pitch (first mode vibration) of the air column in the pipe. On flute this runs from low C or low B to all-open third space C#: on sax from low Bb or low A to all-open third space C#; on clarinet from low E to throat A#.
If you set the air column into second-mode vibration in a saxophone or flute, it sounds an octave higher, and that's what I'm calling "clarion register", what flute players call "the second octave". Now because of the peculiar characteristics of the clarinet, the second-mode vibration isn't available and the first overblown register sounds a twelfth above the fundamental, thrid-mode vibration. On flute this register runs from third line D to all-open second ledger line C#; on sax from third line D to palm key F or palm key F# if so equipped.
Higher modes of vibration of the air colums are called "altissimo" notes. On clarinet it's those above the clarion register, on flute we call it "third octave" though it actually starts with D on the space above the second ledger line; on sax it starts with the fork E. I haven't played enough clarinet lately to be familiar with the transition notes there; I'd have to get mine out and shed on it a bit.
I admit that using "chalumeau" and "clarion" to identify saxophone and flute registers is a bit unusual but it's fully supported by physics and since those designations work well on clarinet I see no reason not to use them for saxophone and flute. Especially on saxophone where altissimo and clarion registers overlap and there seems to be great confusion.