my understanding is that:

• nonbinary: identifies in any way that falls outside the woman/man gender paradigm. this includes but is not limited to agender people, and some other examples are genderfluid, bigender, and xenogender identities
• agender: lacks gender entirely or does not identify with any particular gender

so all agender people are nonbinary, but not all nonbinary people are agender

Some agender people reject the NB and trans label, others use it. Personally, I've used all three but there was a point where I specifically identified as a gender but didn't think trans or NB were useful labels, so I rejected them even though I thought I'd technically fall under the broad definition.

For me, I was attracted to the agender label first because my basically feeling about gender was in was too autistic for gender. Before adopting the agender label, I was confused how someone who was autistic could be cis or trans. I thought gendered restrictions and norms were annoying and felt like I tended to ignore gender more than most on accident (ie: someone else pointing out that I was the only guy in a group on a couple occasions... Once when it was supposed to be girls' turn to change on the bus... Oops). Agender people at least on reddit seem to be overwhelmingly ace.

NB is a big umbrella. It would include agender, bigender, poly gender, demigender, gender fluid, etc.

This is why I tell people "any pronouns" despite almost always referring to myself as "it" or "they". I find it useful to hear how I am percieved, it helps me figure out just how "extra" I need to dress to be seen as a person of mixed gender.

"Binding" refers to the act of compressing one's chest, usually with special clothing.