For the edit, I popped the color by balancing the color, fixing the exposure and saturating.
As for them being "candid"... ya, I guess you could mince words. I trhink childrens portraiture is more about taking a few awesome shots than taking 300 and hoping for the best. It goes with having an idea in mind and executing it- Bringing bubbles, or a chair or prop, etc. Planning, and executing, knowing your light, how to use it, how to interact with the child, and really just having the picture in your mind before hand, and then following through on it.
Its not that strong shadows are less desirable in baby photos, it is about using them appropriately. In 2, its more a lack of eye contact or lack of connection in the photo than anything. Its not interesting. The shadows come second fiddle to that, I think. If I had 1 and 2 in front of me, I would only be showing one of those to the parents , and immediatly I would toss number 2. 1 is strong and the same pose and same setting, same clothes etc. Close enough that I would pick the stronger photo and ditch the other.
Here's one of my son where the light is not directly in his face, but I chose the direction purposefully to get a nice backlighting.
#3 might have worked had you tried to back light him. That shadow of the leaf across his face just seems distractable to me and not really... showing the viewer anything.
As for #4, I can't tell you much about in regards to alternative lighting since I don't use any myself. The bakcground where its wrinkly and weird, might be better cloned out so its uniform. And yes, (for me) I might make the catchlights better in PS. I really love a good catchlight.