In the context of your examples:
- The first has an implied "still": I am still here; I continue to participate.
- The second refers to the location or situation: I could have been somewhere else or in some other situation, but I am here or in this situation, instead.
- The third introduces myself and goes on to describe me.
The three phrases are somewhat ambiguous. "Here I am" could just as well have been used in the first example in terms of meaning, but it sounds more melodious to have the first two sentences mirror the same construction. "Here am I" wouldn't really fit the first example.
"I am here" could have been used for the second example. But "I am here instead of there" sounds a little trivial (answers "where am I?"), while "here I am instead of there" stresses the "here" to provide better focus on the point of the sentence (a comparison between "here" and "there"). Again, "here am I" doesn't really fit this usage.
Both "I am here" and "here I am" could be used in the third example, but "Here am I" is better. That sequence is like a form of presentation or introduction. It introduces me and then goes on to describe me.
There are a number of other ways the three phrases can be used. For example, "I am here" could relate my location (e.g., uttered while pointing to a map or answering the question, "Where are you?"), or announcing the fact that I have arrived.
"Here I am" could be similarly used, but you probably wouldn't use it while pointing to a map because the map location would be an abstraction while "here I am" would refer to your current actual physical location.
"Here am I" isn't typically used in conversation. The only way I've ever seen it used is to present or introduce yourself as if you where someone else doing the honors. If you're familiar with the old TV program, The Tonight Show, the announcer would introduce the host, Johnny Carson, with, "Here's Johnny!". This is similarly introducing yourself, "Here am I."