Welcome to the forum.
"live streaming" usually refers to sending video over the internet.
If what you want to do is record and watch on a projector at the same time, you want a camcorder with an HDMI port a projector with the same. You will project an exact duplicate of what is on the cameras viewfinder screen. When you are looking at camera, you want to find one that has an option to turn off every icon displayed. Having a PC in the loop is superfluous.
In other words, it is common for things like camera settings or the "REC" to be displayed while recording. Some camcorders allow that to be turned off and some not.