Use Canon T3i as a HD Webcam
You can probably do it directly through the EOS software.  It is possible to get a live video stream from the camera over the USB connection.  Note that USB2 is not going to be able to carry a full quality HDMI stream though.  This is why the Intensity is USB3.  It requires USB3 because the bandwidth requirements for the quality level it is working at require USB3 data rates.  If you have a firewire port on both, that would be an alternate option.

There are systems that have HDMI capture, but they are due to having an encoder card like the internal version of the Intensity.  If it is a desktop that doesn't have USB3, it may be possible to get a USB3 card for that system.  I personally have used an addin card for USB3 support on my desktop (a first gen i7) and it works great.

No matter what you do, there is a good chance you are going to have to hack your way to getting it to show up as a webcam.  It would likely show up as a capture device though, which should work with EasyWorship or ProPresenter.  (Personally I'm an old MediaShout fan that is looking at swapping to ProPresenter since Media Shout went pretty awful with version 4, but I've used all 3.)

One other option you haven't considered is that there may be a way to use the USB connection to pull the LiveView feed that EOS Utility uses, but it still wouldn't be as good of quality as something like an Intensity, but free is good and it would likely support USB2.