Is there any software that can capture the screen and turn it into a fake webcam input?

I think, fundamentally, the underlying meaning is the same. The difference is in the emphasis and softening a rebuke in this example, perhaps.

“You know that we have pizza on Thursdays, right?” - is almost a rebuke (without context it sounds that way at least). It is the fact that is significant here - a point is being addressed.

“You know how we have pizza on Thursdays, right?” sounds more like either an excuse is coming or a change of plan. ‘That’ could also signify the change of plan, but it is then more formal/significant.

Whereas dropping the conjunction, “You know we have pizza on Thursdays, right?” sounds more like a mild rebuke, or perhaps a memory jog.

To me, without context, this is what I hear...

“You know that we have pizza on Thursdays, right? Well, in comes Bob with Chinese takeaway. On a Thursday!”

“You know how we have pizza on Thursdays, right? Well, for a change I thought we’d try that new Italian rather than The Hut”

“Your mum is coming on Thursday? You know we have pizza on Thursdays, right?”

here's mine: https://github.com/rdp/screen-capture-recorder-to-video-windows-free it's free and open source

VH Screen capture was exactly what I wanted. It comes in a bundle with VH Media studio which lets you mix multiple Webcams, with Picture in picture and fade effects, which was great for what I was trying to do.

This appears to be another one:

Medialooks Screen Capture DirectShow Filter

Unreal Media Server has an optional add on called Screen Capture DirectShow source filter (half way down the page)

I haven't tested it.

Well. I know FFsplit has a webcam output and works really well. I've been using that for ages, it cannot be started over command line. It can also stream to the major streaming sites and output to video. It supports picture in picture with multiple layers of regions, monitors or windows themselves.