There is no reset button on my icloud.com page!

You can reset the Photo Stream only from an iOS device. Not through a Web browser.

This should work for you from icloud.com... I use it all the time!! Good luck!

Reset your Photo Stream

You can reset your Photo Stream by removing all your Photo Stream photos from iCloud (the last 30 days of photos uploaded from your devices).

Note: Resetting your Photo Stream doesn’t remove any photos already downloaded to Photo Stream on your devices, nor does it remove photos already imported into your iPhoto or Aperture library (if automatic import is turned on). For more information about deleting photos from Photo Stream, please go to this support article on the Apple Support website.

HideReset your Photo Stream

1. On a Mac or Windows computer, go to icloud.com, and sign in using your Apple ID (the one you use with iCloud).If you're already using a different application on icloud.com, click the iCloud button to switch applications.
2. Click your Apple ID account name at the top of the page.
3. Click Advanced, and then click Reset Photo Stream.
4. Click Reset in the confirmation dialog that appears.
5. Click your Apple ID in the upper-left corner to return to the previous screen, then click Done (in the upper right).