So, you always have to specify a channel when pushing with butler, like so:
butler push folder user/game:channel
If your game was previously uploaded via the website, you'll just need to upload it once with butler with a channel name of your choice (that contains 'html5'), delete the old one, and mark the new one as "This file will be played in the browser", and you're all good to go!
The next time you want to upload an upgrade, use the same channel name you just created, and it will update the embed, and distribute patches to itch.io app users. Hope that clarifies things!