A "Bill" is the shortened term for any number of weapons based on the billhook, a forward-curving blade used much like a machete, for chopping and clearing heavy vegetation. While the weapon form is/was often seen as a pole-arm not entirely dissimilar to a halberd (with a hook in place of the axe head), a bill could also be a hand weapon similar in size to an axe or short sword. The bill described in your diary entry is likely a basic agricultural/landscaping implement (still available today!), and like some machetes, isn't overly sharp due to relying largely on the force of the blow and the favorable shape to do its job. Too sharp an edge is very vulnerable if you're going to be hacking at tree limbs.

There are various versions of bills, with various name applied. I can't speak to a "backed bill" in particular, but there are a number of variants that have a reinforced blade on part of the back, as well as the regular hook blade on the front. I would suggest pursuing that line of research.

A pimento stick would refer to a walking stick made of pimento wood. A rare (at that time, anyway) tree found almost exclusively in Jamaica, and the source of allspice.

Sources: Bill as weapon: "Medieval Weapons: An Illustrated History of Their Impact" Images of Billhooks:Google Images Random example of modern bills for sale: Machete Specialists Pimento Trees: [From Floridata] (http://www.floridata.com/ref/P/pime_dio.cfm)

The English bill is a pole arm which first came into use in the 17th century. Similar in form to a halberd, It was first a modified farming tool known as a billhook before being adapted and manufactured specifically as a military weapon. I would assume this entry refers to a billhook with a shorter handle which could be carried and possibly used effectively as a defensive weapon if sharpened beyond what is necessary for its farm duties.

Weapons of the 17th Century - Chatura Weliwitigoda