The way I do it is shown in the bottom of the 1st photo. I knit to the thumb opening, put the thumb stitches on waste yarn using a tapestry needle, and cast on the same number of thumb stitches using the backwards loop cast on. (If I'm doing colorwork mittens, I cast on using both yarns in pattern.) I prefer this method for two reason - it allows you to try the mittens on as you go to make sure they're the right length and I think it is easier to pick up stitches in pattern when knitting with 2 colors. While the first method is quite popular with knitters actually the second method of dealing with a mitten thumb is called the traditional method in Lizbeth Upitis' Latvian Mittens book.
If you do your thumb the first way, be forewarned that you will not get as many live stitches on the back of the thumb opening as on the front when you remove the waste yarn. This can screw people up in their stitch counts. The only place I've ever seen this phenomenon mentioned is Nancy Bush's Folk Knitting in Estonia. In the photo below after the waste yarn was removed there are 8 live stitches on the front of the thumb opening but only 7 stitches on the back of the thumb opening. (The missing mystery stitch will be a half stitch on the upper right hand side of the opening in the photo.)
My goal is always to make the colorwork pattern match up on the inside of the thumb as shown below to keep the feeling of a true invisible thumb. Of course this isn't always possible if the pattern itself doesn't give the right number of stitches to keep the colorwork pattern seamless.
I'm off to see if I can get some good photos of how I pick up the stitches.