1090MHz ADS-B messages are part of Mode-S messages. To be precise, part of Mode-S downlink messages. Mode-S messages include all types of messages that Mode-S transponder would transmit, in which there’re 1090MHz ADS-B messages. UAT ADS-B is a totally different system and has nothing to do with Mode-S. Mode-S messages are explained in ICAO Annex 10 Volume IV and ICAO Doc 9871. Mode-S messages include all types of Downlink Format, shortly as DF. As DF0, DF4, DF5, DF11, DF16, DF17, DF18, DF19, DF20, DF21, DF24, different types messages. In these types of messages, DF17, DF18 and DF19 are ADS-B messages. To better understand ADS-B messages, can read RTCA DO-260 series and EUROCAE ED-102 series documents.
One more word, this is why ADS-B receivers can also receive Mode-S messages, and Mode A/C messages.