No, a non-folding Minivelo, in part or whole, is most definitely not carry-on luggage, even here in lax Asia-Pacific.
And btw, non-folding bikes are not allowed on the AREX train out of Seoul's Incheon airport (which is actually quite far from Seoul off-shore on an island). As soon as I approached the train turnstile, I was accosted by the station manager who emphatically said no to my 16" wheel unfolded folding bike. So, what did I do? I folded it, was let in, and got on the AREX.
If you are flying into Busan's Gimhae airport, that's another matter, for you can literally ride from the airport to the city in less than an hour.
Riding from Busan to Gimhae airport
You can ride to Seoul from Incheon airport too, but that will take you at least half a day. Or put the bike on a bus.