First category comes in numerous designes and configurations. Perhaps this is the most common form of the speaker.
The advantage of this speaker arrangement is that by using two speakers one can get much better extension on both ends of frequency range, specialized speakers for smaller frequency range have usualy flatter frequency response, decreased distortion and increased power handling.
The disagvantage of this speaker arrangement is introduction of passive crossover, although biamplification with active crossover can be done too, there is still a crossover with its limitations and compromises. Another issue is the use of two speakers as two point sources for sound, which leads to phase shifts, differences in horizontal vs vertical frequency response and so on. This can be partly avoided by the use of coaxial drivers.
All the enclosure types in order to extend low frequency range as mention in one way speaker systems apply here as well.
another coaxial, Tannoy
with horn tweeter
line array, Dali
Second and quite different approach to two way speakers is the use of full-range speaker supported with subwoofer. In this arrangement we have all the advantages of full-range speaker, no crossover in the most critical region of frequency spectra, plus a good bass extension. Various combinations of speaker enclosures can be used.
if you are very brave with waf or live alone, you can construct this industrial looking two way system with full-range accompanied with subwoofer done as horn with compression driver, bravooo...