That "TheWorld" browser link does not work for me.
Tumblr video refuses to play in K-Meleon, Pale Moon, or Firefox. The play button does nothing. (My Pale Moon's version of Goanna is similar to Gecko 38.8 I believe)
I tried Tumblr video in the latest version of Opera however (36.0.2130.75; based on Chromium 49.0.2623.112 it looks like) and it tells me right out that "No compatible source was found for this video". On Twitter in Opera, I get the same old "This browser does not support video playback." message as I go in the Gecko/Goanna browsers. Opera does seem remarkably fast and lightweight on my system however!
The customization options leave something to be desired (as in any Chrome clone, just like Mozilla Firefox for example!)

All this on Windows XP SP3 with latest updates for SP3! (not for POS-Ready 2009!)

I just tried the demo on http://www.w3schools.com/TAgs/tryit.asp?filename=tryhtml5_video, and it works great with OGG Vorbis video, but when that line is removed it cannot play the MP4 video.

This seems to be a OS problem, not a Gecko problem since I get the same result in Chromium-based Opera.

Has anyone found success with using an updated version of the K-Lite codec pack??

This isn't OS problem (unless you aren't under XP) - this is engine-related problem.

Only new Geckos have ability to play such videos.

Well it seems to be related to running XP or earlier OSes.
I guess it must be a Gecko, Goanna, and Blink problem then!