Whether to start at thumb root or to make a separate thumb
The thumb can be either started at the edge of the thumb opening, or you can first make a separate thumb, and attach it by sewing by hand. The first mentioned is more common.
Video - How to start the thumb, corners, finishing off
Video - Thumb placement in a square top mitten *) read more at the bottom
Video - How to handle the unwanted gaps at thrumb root corners
Whether to start at the upper or bottom edge of the thumb opening
You can start the thumb at either edge of the thumb opening, upper or bottom. I guess it is up to you, how you prefer to do it. Some people say they start at the upper edge, so that the first, sloped row won't show. Some others say that if you start the thumb at the upper edge, the thumb will flop downwards. I have usually started at the bottom edge, and I don't think the first sloped row has bothered me too much.
In addition, in at least one book advices to do the first row only partially: Start at the upper edge, near the corner, and continue to the bottom edge, and back to the upper edge around the other corner. The first row is sort of C-shaped. Then, you'll restart at the upper edge.
You can leave the yarn tail a bit longer, and you can afterwards use the tail for darning if there are gaps at the thumb root corners.
How to start the thumb
You can start the thumb by first making a small loop, and then increasing the stitch size little by little. Video (linkki)
There are several ways how to start the thumb, and one of them is shown in Soisalon-Soininen's article (in Kotiteollisuus magazine 1956) with Russian Stitch. Wrap the yarn around finger three times (link), and then pick up the connection stitch(es) at the edge, and one or two loops at the corner of the thumb opening, and then insert the needle through the yarn loops under-over-under - turn needle - over-under over. Then do another stitch by picking up an edge loop (or two), and again insert the needle through the yarn loops under-over, according to the stitch type you are using.
Another way to start the thumb is to make first a knotted loop like shown here (link). Pick up the connection stitch(es) at the edge, and one or two loops at the corner of the thumb opening. You have the loop (with the knot behind) around your thumb, twist the needle around the loop behind the thumb, and then continue working as usually.
Corners of the thumb opening
At the corners of the thumb opening there aren't actual edge loops for connection stitches, but usually you can make a stitch or two at the corners, too, and connect them to where you best can connect them. Usually it is good to make the connection stitch a bit 'deeper' than just to the very edge. If it seems that there will be a gap at the corner, you can undo the stitch, and try again with a bit different connection. You can also darn the gaps afterwards.
Continue nalbinding until your thumb is almost covered. Then start decreasing in every stitch, and keep on decreasing until it is difficult to pick up the stitches, or the opening is only about the size of a finger tip. Pull the last few stitches small carefully by hand in order to avoid loose loops. Either sew the gap closed with a couple of stitches, or insert the needle (a thin darning needle works better) through all the last loops, and then pull the yarn to close the gap, and secure with a couple of stitches. Hide the yarn onto the reverse side by weaving it into the loops there.
If you are making square top mittens, and want to make a square top thumb, too, instead of decreasing at the top, you can close the opening at the top by sewing by hand with invisible stitches. Hide the yarn onto the reverse side by weaving it into the loops there.
*) Note. If the mitten shape is about equally wide all through, and at the thumb opening there are as many loops at both upper and bottom edge, or only a stitch or two more at the bottom edge, the top part and sideline of the mitten twist when you are wearing the mitten. To avoid this, you can place the thumb opening a bit across the sideline, like shown on the video. On the other hand, if your thumb opening has plenty of more stitches at the bottom edge, you can place the thumb opening at the side. In the round top mitten it doesn't matter where you place the thumb.