The Philips that works allows for the following resolutions (from the owner's manual):
640x480@60Hz ; 800x600@60Hz; 1024x768@60Hz or the video formats 720p and 1080i.
The Sony that stops working allows for:
576i, 480p, 576p, 1080i and 720p
The Sammy 32f4000 works with: 720p (and I couldn't find anything else)
The Sammy ue40es6100 seems to work with 720p, 1080i and 1080p.
Your Toshiba laptop has a native resolution of 1366x768.
So, my guess is the older Philips is able to handle the not-so-standard 1366x768 resolution of the laptop while the rest of the TVs do not.
Here's what I would do: I'd set the laptop resolution to 1280x720 (720p) and hopefully the laptop screen can handle that. I'd then verify I was in clone mode for the external monitor (in other words the external monitor resolution is the same as the built-in LCD). I'd then try the Sony, Samsung and Samsung again to see if 720p works.
That R-Pi adapter is interesting. I doubt it will be on the market for long since it claims to handle HDCP. It is actually much more than a cable.