AVC1 is just H264[0]. FFmpeg says stream 0 is "Video: h264"
The page you linked about AV1 in mp4 says it uses MPEG-4 Part 31, which is under development and not final [1]. Until now I was not aware they were extending the MPEG-4 container standard to include more formats.
Maybe, but not exclusively
VP9: https://www.webmproject.org/vp9/mp4/ (uses 'vp09' & 'vpcC' in the stsd)
AV1: https://aomedia.org/wp-content/uploads/2018/09/AV1-ISO-Base-... (uses 'av01' & 'av1C' in the stsd)
YouTube's AV1 playlist https://www.youtube.com/watch?v=2nXYbGmF3_Q&list=PLyqf6gJt7K...
uses mp4 as the container for the av1 (av01.0.05M.08), webm for the VP9 (vp9)