The strange deformation is because those vertices are weighted to other bones. I guarantee it. Click on each bone and look at the model. The one that is causing the trouble will show weight (white) across the faces. If you don't see any white on those faces no matter which bone you choose then it means that 2 or more bones have weight on that area and each has such a small amount of weight that it doesn't show up as white but it is still there. You can choose each bone and either paint zero weight across the faces or you can just choose each bone and flood the model with zero.

The thing about skin weights is that each weight (the weight on each vertices) has to always equal 1. This means that the weight is normalized. If you paint some weight on an area, the weight is being taken away from one of the other bones that is weighting that area so that the weight always equals 1. Example : Vertices 1 is being weighted by bone 1, bone 2 and bone 3. bone 1 has .25 weight, bone 2 has .5 weight, bone 3 has .25 weight. All together these 3 bones have weight that equals 1.

When working with weights, it is best to "hold" the bones that you don't want to disturb. That way, when you paint the weights, Maya won't take any weight away from the bones that are on "hold" and if you subtract weight from a bone, Maya won't add the weight to a bone that is on "hold". I like to put all bones on hold then selectively unhold the ones I'm working with at the moment. It can be tricky and it takes some practice but you can do it!

About your naming problem. The picture you added is hard to read but, the two bones that appear to be named the same might have a small difference such as a capital letter. Check that out. ... Maya should not allow 2 bones with the same name unless there is some kind of a difference. Maybe each of the 2 bones is in a different group or maybe you changed the name of one of the bones and the old name is still part of the bone's history. Look in your outliner and find the 2 bones then rename one of them to see what happens. Maybe you need to unbind your model then bind it again to see if the problem goes away. Could you have possibly bound the same bone more than once? I don't know if this is possible but you can check.