I wasn't able to find the exact staple gun used in the StretchRelief pliers video, but I didn't want to come back to you "empty-handed." I've done a bit of research on the subject, and here are some key points I've found:
1. Dependent on volume, you may want to consider a pneumatic setup--versus electric--as it is not uncommon for electric guns to have trouble sending staples all the way into the wood frame (air pressure can be regulated and "fine tuned"). The type of wood and its softness/hardness should be taken into consideration here. If you're using a soft wood like pine or fir then you would probably be fine with an electric gun, where as a hard wood like bass may warrant further consideration of a pneumatic setup.
2. When selecting a gun (and the staples for that gun afterward), the "sweet spot" for minimal canvas tearing is a narrow crown staple, where the crown measures between 3/8" and 1/2" wide and between 1/4" and 3/8" long.
3. When longevity is a high priority, be mindful of your staples' material. Stainless steel construction is ideal to avoid rust, though they can be a bit more costly.
Here are a few electric staple guns that I found to generally have good reviews for this type of application:
I hope this information has been of use to you. Please let us know if there are any other questions we may answer for you.