It seems that you want to know how to enable the Closed Captioning feature while using the web based application.
This feature is not related to the Slingbox in anyway, since it is provided by the cable box or DVR you use. In all case, I recommend you to check the following thread in order to get some helpful information:
Make sure to follow the included instructions and let us know how it works for you.
The Closed Captioning thread you provided does not help. Closed Captioning is channelled via the cable and can be view by a device such as a TV with options. SlingPlayer for Windows had CC as a feature to turn off and on. Slingplayer for Windows was replaced by the web-based Sling player. As the web-based Sling player is now the only options for windows I would expect CC be an option there as well.
You are absolutely right, they upgraded technologies but actually downgraded on features!!!
The good thing though is that you can still download the SlingPlayer for Windows, here is the link:
Does anyone know if CC works on SlingPlayer for conected devices like the boxee box?