Stereo 3D image with text

I'm not sure what you are trying to do. The two videos you posted are for turning a flat (2D) photo into a stereoscopic image. AKA, stereo conversion. What I think you want to do is add 3D text to a stereo image or video.

There are many ways to add stereo text to a stereo image. Any 3D graphics program, such as Blender, can generate 3D text (usually a 2D font extruded into the 3rd dimension) which can be easily layered over either a 2D image, or placed into a 3D scene.

"More specific is there a way to create the illusion of depth with say white text on a black background?"

If you just want 2D text to appear at a given depth in a stereo image, then all you need to do is offset the text slightly in the left or right image. You could just experiment with the amount to offset the left/right, or calculate the geometry of your video scene and position the text at a specific depth. It would be then layered over the video in the final render by your video editing software.

If you want to create text as 3D objects with depth, Blender could do that easily.
You can also precisely position the text in a 3D space with Blender's add-in stereoscopic camera.

Tutorial for making 3D text in Blender:
https://www.youtube.com/watch?v=gYXSo2xDUPE

The Blender stereoscopic camera setup:
http://www.noeol.de/s3d/

"The idea is to create an information screen or menu for a 180 degree 3D video"

A menu like that would essentially be a 2D object (flat plane) in the 3D space. The text wouldn't need to be 3D, just 2D on a plane in the 3D space. Again, you could use a variety of software to produce the menu as an image, offset in the left/right stereo view to a desired depth, then layer that over your video in the final render.

If you intend the menu to be interactive, then it will depend on your application for viewing the video, which should have an API that describes how to make an object interactive. If not interactive, just information, then again it is a matter of layering the object (whether 2D or 3D) into both the left and right video, offset slightly to the left or right in the left/right videos, to position it in the 3D space.

Thanks for the great reply. I knew I wasnt entirely clear, Im struggling with the correct terms and still learning about stereo 3D in general.

Indeed the only thing I want is 2D text in a 3D space, non-interactive, pure informational.

Can you help me a bit in the right direction to calculate the offset, just so I understand the concept and can apply it to different video resolutions.

To give even a general notion of how much to offset, someone would need to know the geometry of your videography. What was the camera separation? What was the focal length? How close are the objects in the video scene where you want the text to appear? It gets complex, and I'm not expert enough to figure it out in this discussion. But here are some general pointers.

One is that you need to render the text using the same lenses. You mention that it is 180 degree video, so I presume that means you didn't shoot with "normal" (50mm equivalent) lenses on your stereo camera rig. I would recommend using Blender or a similar 3D rendering app, that allows you to use a virtual stereo camera that can match the properties of your physical lenses. Then render the text as a still image (PNG with an alpha channel) in the same aspect ratio as your video, and layer it onto the final render in your video editing app. Whether you render two separate (left and right) or a combined double image depends on the format of your video.

The blender image examples in this thread might give you some ideas. They are stereoscopic still images for viewing on the Rift, that have text, just as you describe.
viewtopic.php?f=33&t=17444

I was reading along in that topic, however that's not what I want.

I want the text to be flat (2D) in a 3D solid color space. Furthermore, I'm indeed intending to shoot 180 S3D videos with a lens separation of 65mm.

Let's take an example, if I would take the following image of the Oculus Connect logo + text:

https://s3.amazonaws.com/static.oculus.com/website/2014/07/connect_logo.jpg

Save it twice (Left & Right) and then edit the horizontal convergence of the Left image in my video editor, would that be enough to create the depth?

Unfortunately I can't test this atm.