Alright, but still, if a piece changes it's key, is it in the key it started in, or the key it ends in?
Progressive Tonality is the term for that. Karl Neilson, among others, is known for it.
"I can not think of a piece of this nature where the composer stated the key, if you think on it one sec, for obvious reasons, lol."
Naming a piece and stating the key is such a common practice bit of business, and embedded in the general consciousness, that causes, imo, any number of amateur and beginning composers to still think in that mode, for both the actual writing and title. It is a mental reflex, maybe some think your piece sounds more 'official' or legitimate with that old tag on it, but we all know that just ain't so.
Progressive Harmony, shifting key centers without returning to 'home base,' can happen within a movement (your Etude) or within the key schematic of multiple movement works.
Stravinsky's
Piano Sonata has no key name, and its three movements are in: 1 C / 2 Ab /3 beginning in and ending in E major. The first movement C is in the center of its upper and lower major thirds, outlining an augmented triad.
Ditto Barber's piano concerto, with its schematic a sequence of descending minor thirds:
1st movement beginning and ending in E minor,
2nd movement primarily in C# minor (the relative minor -- descending -- from E)
3rd movement "mostly" in Bb minor (polite use of the enharmonic for A#) that again a minor third lower than the second movement.
Call it "Etude," or if you will,
Étude :-) It only needs a number if there are others; will get a number if others come later.
P.s. Nitpick, but its the literal truth:
"I've written an Etude.... If I write it down...."
Well, you haven't written anything until, ya know, it is written down [memory and midi or other recordings don't count as 'written' lol].
Best regards