What is the meaning of the word "bill"?

Definitions:

a brim that projects to the front to shade the eyes noun
- They pulled down the bill of their cap and trudged ahead.
a long-handled saw with a curved blade noun
- The gardener used a bill to prune branches off of the tree.
a statute in draft before it becomes law noun
- they held a public hearing on the bill
an itemized statement of money owed for goods shipped or services rendered noun
- I paid my bill at the restaurant.
to advertise especially by posters or placards verb
- They were billed as the greatest tenor since Caruso.
to demand payment verb
- They billed us for their services.
to publicize or announce by placards verb
- We need to bill this show all over town.