AGOL uses UTC time by default. Collector will convert the date/time using the computer or phone's date settings. But to correct the date and time in the downloaded data, you will need to add/subtract your time offset from UTC. Search AGOL or ArcMap help documentation for "geodatabase time".
You also indicated that the field name is "DATE". Field names should not use SQL reserved words. If you want it to show up in Collector as "DATE", you can use the description/alias for this purpose.
