What equipment can I use to record presenter's computer screen?
Yes. In the old days, we would use a scan converter which took a computer video signal and output it as S-Video. 

You have a few options. The least intrusive is to continue to use your existing method of recording the screen, but to request the slides from the presenters.  Then, using the video recording of the screen for timing, you overlay the slides at the appropriate times. If you save the presentation as a website in Microsoft PowerPoint, it will generate a series of JPEG images which are handy for your video editing.

Other methods require you to have access to the computer or the connection to the projector. If you can install software on the presentation computer, you could do a video screen capture.

If you need to capture animations or video, you'll need to capture the actual video signal from the computer. You could send the computer output to a distribution amplifier, and send one of those outputs to a recorder, or you could install an SDI video card, and send the output to a recorder. This is the highest quality solution, but probably exceeds your budget.