I pushed this branch to one of the same name for this repo, it generated https://hub.docker.com/layers/247489278/anacrolix/dms/add-docker-publish-image-action/images/sha256-3c9fdb05e699f5973870f248944b516f03d19fc982082793512674da0d166f88?context=repo. Looks good? Do I need to do anything to get it to generate previous tags, or does it just do it going forward? When I moved/merge this with master, I'm guessing a master image will be generated?

The image looks good. I've done a cursory run on my PC, looks good. As soon as you merge it, :mastertag will get pushed.
I'm not sure, but maybe the SemVer image tags don't generate until a new version git tag is pushed, no previous tags. Just pushing a new release git tag would generate :1, :1.x, :1.x.yand :latest image tags (the latter is expected as a standard practice).
From the top of my head, no way of generating previous tags, AFAIK. Some sort of deleting tags on Github and re-pushing them (at the same commit ids) come to mind, but it has obvious caveats. If I come with any solution I'll let you know.

For reference, check out my test image tags (pay attention to image hashes to see the same image pushed to multiple tags).

@anacrolix Keep in mind that until a tagged version is pushed, the only usable image tag is :master (which is not default), and therefore any image run or pull without specifying that tag will fail, as :latest is always assumed if no tag provided, as explained above.

$ docker run --rm -it anacrolix/dms
Unable to find image 'anacrolix/dms:latest' locally
docker: Error response from daemon: manifest for anacrolix/dms:latest not found: manifest unknown: manifest unknown.
See 'docker run --help'.

I've just pushed v1.5.0. Let me know if it doesn't work out!