I can see how they are both similar though. They both build up to and introduce a haiku  with prose and then are followed by a haiku. 

I feel like I might have to be more involved in the moment, or like the moment that I’m writing the prose and the moment I wrote or feel inspired to write the haiku ought to be connected, but maybe that doesn’t have to be the case.

A haibun is a combination of prose and haiku. The word literally means "haikai writing", haikai being an old word for haiku. The prose could be anything: fiction, non-fiction, personal writing, etc. Usually though it is a bit longer than just a few lines. A few lines before a haiku is often considered just a headnote. Almost all haibun I've read have been at least a few paragraphs. 

The wikipedia writeup is a little dry, but fairly good: <https://en.wikipedia.org/wiki/Haibun>

That said, there is no clear definition. One of my favorite haiku poets, Santōka, for instance, would write long diary entries surrounding all his haiku, but he didn't call them haibun so we usually don't consider them haibun. 

When I think of haibun, I always think first of Bashō's *Oku no Hosomichi*, which is usually translated to something like "The Narrow Road to Oku" or "The Narrow to the Deep North, and I think most Japanese would instantly think of this book first. It is a collection of travel notes about areas he visited during his trip and then a haiku about each place. (You can find the entire thing [here](https://web.archive.org/web/20090201143706/http://darkwing.uoregon.edu/~kohl/basho/1-prologue/trans-mccullough.html) including a few translations of each section if you want).

The difference may be that Bashō's book had an ongoing narrative, whereas Santōka's writing was more isolated into sections. But then again, one could argue if Santōka published any of the sections separately, he could have called it a haibun. So again, it's a little bit of a fuzzy term. 

I think what @boxcarblue writes here could go either way. If he called it a haibun, I wouldn't argue.