hub-and-spoke v. point-to-point
With the blurring of the boundaries between legacy and no-frills, there is only one fail-safe way to tell the difference: the operation of a hub-and -spoke v. pure point-to-point.
Legacy carriers provide interlining and through ticketing on their own flights, on those of it's "parner" airlines, and competitors in general, and baggage can be checked through to the final destination.
This is not the case on no-frills operators where two connecting flights, even on the same carrier, are two separate contracts, and baggage has to be collected and then dropped off again (hence no protection for missed connections, etc.).