Not much you can do about that. Red is a particularly nasty color for video files. The more saturated the red is, the worse it gets. You can TRY masking around the red text and using the hue/saturation effect to reduce the red saturation, but I don't think it's going to sharpen up the text -- it was recorded that way.
Hmm, that's interesting.
In my case, yellow, white or blue (on player's name on video) all look almost perfect but not red?
Red saturation is always that bad for video recorders?
Since my monitor is 1080p and when watching this 720p video, this "red" color looks more worse.
Especially worse on YouTube.
The irony is that when I record the game from a phone (samsung galaxy s3) in 1080p, the red text looks perfect - as in not saturated as white, blue or yellow. Even look the same on YouTube.
Hmm, there must be a solution...
I can't see your images behind my company firewall. Different devices ask for and are served up clips using different codecs from the online servers. It's not surprising at all that the colors look different on different devices, video is not a constant. The problem with super saturated colors is they tend to confuse encoders during the compression and decompression cycles. This artifact is a reality of compression technologies but it can be mitigated with careful planning. You can put a border around red text. You can reduce the saturation or, using a vectorscope, blend a bit of green and blue into the red to move the vector dot from the pure red box. Game video presentations are designed for maximum visual impact usually by people who have no understanding of or regard for video standards and conventions which are in place so video clips can be passed around between various platforms and systems and remain more or less intact and viewable. You can compare this problem to audio systems. If you have a 5.1 track that is meant to be played on a huge system, how do you make that available or at least discernible for efficient playback and enjoyment on a phone?
Since you're the one making the recordings, here's my suggestion: not all screen capture applications are equal. Get a better one, or change the capture settings on the one you have now to record better video.