How do I wrap text around a cylinder in Illustrator?

To wrap text around a cylinder in Illustrator you need to start by creating a cylinder. (Make it a color that will contrast nicely with your text color)

• Create a smallish circle:

  

Beginning with a smaller circle will make a narrower cylinder with less extrusion necessary, and the text will wrap more obviously than on a thick cylinder.

• With the circle selected go to Effect > 3D > Extrude & Bevel:

  

• Turn on Preview (bottom left) and crank up the Extrusion Depth as desired… (I started with a 140px circle and extruded to a depth of 650) then click on ‘More Options’:

  

• Crank up the Blend Steps (I entered the maximum of 256) and click on ‘Map Art’

• First for the Surface choose ‘3 of 3’, then choose your symbol. You’ll have to scale and rotate as needed. (You may or may not want to add shading. Your example has none so I chose none):

  

• Click OK twice

Lastly, go into the Appearance Panel and lower the Fill opacity of the ellipse so that only the text is visible to match your example:

Now you can just click on the 3D Extrude & Bevel effect and modify the rotation on the Y-axis to get the appearance you’re after:

Final result:

EDIT: Just looked over your question again and I realized I somehow missed the link you provided to the full-size example. I didn’t realize you were looking for almost a dozen different versions rotated to varying degrees…

To get the iterations as viewed from the back (as the side of the cylinder is rotated away) when you’re in the Map Art dialog make sure to check the box for ‘Invisible Geometry’:

In the end though – if you’re a stickler like me – then the real answer is:

Your example was almost certainly created in After Effects. Here I’ve recreated it in AE and you can see the perspective matches your example far better than the Illustrator method: