Good question. Customer has paid for planning work and has verbally
agreed.
Red flag #1: verbal agreements are worth the paper they’re printed on.
That said, if they’ve paid for planning work I guess the natural deliverable for that would be a plan of some sort. So let’s address that. It sounds like you’re still in danger of being shopped with your own document, but…hazards of the job.
Now we want to agree exactly what the user will see and what will
happen when they do stuff.
That, my friend, is called a functional spec. Now you’re in my wheelhouse. 
Still need to know this:
Do you have Photoshop mock-ups of the UI? Do you have a mood board? Or
do you mainly have those workflow & use cases figured out?
At a bare minimum, I’d recommend the following:
-
Bang out some simple wireframes using any tool that lets you draw rectangles. Hell, you could even pull this off using MS Paint (although I don’t recommend it!). At this point, simple rectangles bereft of any design-y elements are the way to go (you want the customer focused on how it works, not what font you’re using). If Visio is too much to deal with, try something like Gliffy, which is online, excellent, and free.
-
Slap 1 wireframe on each page and include some notes as to how that page of the site will work.
-
If you have use cases or user stories, be sure to include them in step #2 along with your notes.
-
If you don’t have use cases or user stories, write some!
-
Maybe slap a whole flowchart (like the one you hand-wrote) early on in the document, to provide contest for what follows.
Google “Painless Functional Specifications” and download the document Spolsky came up with for whattimeisit.com, or maybe even the Project Aardvark spec (although that might be a bit much for what you’re doing).
Note that this is a bare-minimum approach, if I knew more about your circumstances I’d suggest something a bit more fleshed-out.
