Your camera is acting normally as it was designed to do. The "issues" with which you are concerned are all explained in the [*EOS Rebel T3i/EOS 600D Instruction Manual*][1].

> So, I only should be able to use auto shoot all 3 images using 2 second timer? In that mode, it waits two seconds after shutter button being pushed, and then takes all 3 images. But I have seen video's of same camera doing it in continuous mode too.

The EOS Rebels are all set up this way. So are most, if not all, of the other EOS cameras. There are cameras from other makers that can be set up to take all three bracketed shots with a single shutter push, but Canon cameras don't allow the option.¹

In the videos you are watching either the shutter is being held down in 'Continuous' drive mode, a wired remote with a built-in intervalometer set to take a longish exposure (say, two or three seconds) is being pressed once and sending a "shutter button down" signal to the camera for several seconds with the camera in continuous drive mode, or a different camera is being used in the videos.

It might be possible to do what you want by setting an RC-1 or RC-6 wireless remote (but *not* the RC-5, which can *only* be used in 2-second delay mode) to the "zero delay" setting so that the first photo is taken immediately when the button is pressed with the camera in 'Self Timer (10 second)' mode.

<sub>¹ If there are any exceptions it would only be with some 1-Series bodies or possibly the 5Ds/5Ds R or 5D Mark IV. Both my 5D Mark III and 7D Mark II, as well as my older 5D Mark II, 7D, 50D, and Rebel XTi/400D are all the same as your Rebel T3i/600D in this respect.</sub>


  [1]: https://media.the-digital-picture.com/Owners-Manuals/Canon-EOS-Rebel-T3i-600D-Digital-SLR-Camera.pdf