Support

You have linked to the same screenshot twice. However, that's not important.

The video was produced when Dropbox was using an older version of its API for third party developers. The new version of the Dropbox API has the second button in a different page, due to the massively different way the new API works. You just need to click on the Step 1 button, allow Akeeba Backup to link to Dropbox and then you see a page with the token. That page has a button which you are instructed to click to copy the information to Akeeba Backup. Doing that the authentication window also closes. There is nothing else to do. Just Save & Close the configuration. It seems that you've already done that so there's nothing else to do.

The other problem you have is unrelated. Please do remember that the upload to Dropbox (or wherever else you've configured) is only performed at the end of the backup process. Since you have a problem with the start of the backup process this is unrelated.

According to the log file there is no server-side issue. The initialization returns successfully and your browser should, at this point, execute the next step. Do you get any message on your screen? If so, please do provide a screenshot. If you don't, please try disabling all browser extensions and all software which could interfere with Internet communications. Alternatively, try using a different browser. If you have any smartphone which became available from 2012 onwards you should be able to use it to run the backup. This will let you check whether it's something in your browser / computer getting in the way.