If you ask for an official answer, it's likely to be 'if it's a bike, you have to pay extra'. But as Jim says, how would they know? A suitcase is a suitcase. Unless it's overweight or oversize, it's fine.
A regular case that fits on the check-in conveyor will be fine. If you're asked, it's clothes and bike parts. In reality, they'll ask (or you have to verify at online check-in) whether it contains explosives, fuel, matches etc etc. The answer's no. (I hope).
The excuse for charging extra for bikes is that they require extra and different handling from regular, suitcase-sized luggage. They won't go through the regular conveyor system, so they need to go through the oversize baggage channel or be swabbed for explosive residue at extra cost to the airline. They can't be stacked with regular luggage and are awkward to handle. They may not fit in those wedge-shaped luggage containers and so may have to be slotted into the plane hold separately.
If your bike is inside a suitcase-sized bag and isn't overweight, the only people who know or need to know what's in the bag are you and the person who looks at the X-ray - and they don't care what it is as long as it isn't likely to damage other luggage (although they're not usually fussed even about that), catch fire, or blow up the plane.
There are other good reasons for packing a bag in a hard suitcase rather than checking it in as a bike, besides those above. Over-size luggage is often last to arrive in the baggage hall at the other end, and sometimes its very unclear where you should collect it from if it doesn't arrive on the conveyor. A bike in a bike box or even a bike case can be very awkward to transport to and through an airport - it won't fit in many taxis, often won't fit on a luggage trolley and it may be difficult to get it in a lift (if you can find one) or through security doors (the kind that only allow one person through at a time). I've had bikes disappear for hours because, not being part of baggage handlers' regular routine, they were taken off the plane and stood to one side while they loaded up the regular bags, intending to put the bikes on top - but then forgotten. Finally a well-packed bike inside a good hard suitcase is pretty much invulnerable to baggage-handler abuse, unlike a bike in a soft bag or box. (Yes, a hard bike case is tough too, but being oversize, still subject to the other disadvantages).
I've travelled many times with a Moulton AM dismantled in a hard case or a Brompton in its (suitcase-sized) B bag, and quite a few times with a regular bike in a plastic bag, with its handlebars turned, pedals off and so on.
A suitcase-sized bag is great if your bike can be made to fit; the next-best option has been the bagged bike as a bike.
(Sometimes, check-in staff (if they know how to deal with it at all), insist that the bike must be in a bag. "Okay", I say, "I'll wheel it as-is to the oversize luggage channel, and then bag it up there" - and usually that satisfies them. The oversize luggage guy then usually says, as long as it's padded up with pipe lagging and the bars are sideways and the pedals removed, leave it as it is, then we can wheel it around easily, rather than having to carry it, and it's easier to pick up if you can grasp the frame tubes: you can't if it's in a box or bag. And from my point of view, if you can see that it's a bike, then, maybe, if you're lucky, handlers will treat it like they might their own bike, with a little finesse - unlike the way they tend to lump around cases).
