Record Screencast of Remote Presenter
You have two options, one, install software on the presenter's system that can encode and send a video feed to another system.  This is in-effect going to be a screen casting application, even if the feed is only forwarded to another system to be packaged.

The other option is to actually capture the raw output from the presenter's system through a frame grabber or video interface of some type.  This is the least invasive way, but then have no idea about cursor position or the like.

There is no solution that can connect to another computer and grab whatever is on their screen without putting software on that computer.  You can however do something minimally invasive like putting a VNC server that can run from a USB stick without installing and then connect to that VNC server and run the screen capture off the VNC client on a system you control.