Remember that a board's width is only the widest point; if you have a longboard shape with parallel rails, then it will be wide for quite a lot of the board. If you have a hybrid or egg shape, then the wide point may literally be a single point.
So why the hired board was easier to turn could well be the outline rather than the width (given you say it felt a lot more narrow along the front). Narrow nosed boards can be easier to paddle, especially through choppy water; although if you are paddling a long distance then that advantage disappears in my experience (the broader/more stable nose will help out). But just paddling from shore to the waves, narrower noses can be easier unless conditions are glassy. They cut through the water/waves rather than just slap up against them.
As to catching the wave, that may be related to paddling but it just might be a wider or thicker tail, or you were 'on' with a new board because you were really focused and not being lazy. Although less stable boards feel faster because you are doing a lot more with your body; stable boards you stand there and can just take it easy. All else being equal narrower boards will be less stable; but a narrow nosed/tailed wide board (eg hybrid) will be less stable than a thinner board with a wide nose
Narrower boards wont be faster, but they will be easier to go from rail to rail simply because there is less distance to travel. That can cause them to create a lot of speed. I like this article, saying 'narrow boards are quicker, wide boards are faster'
http://www.surfline.com/blogs/talking-d ... s-a_40518/ But the outline of a board is a pretty minor factor in speed. Think of fins, rocker (especially), concaves, rails, tail size - all of which have more of an impact than outline.
So, as always, its complicated. If you liked the board you hired, then by all means think of getting a new board. Plenty of people have boards that are almost the same dimensions and one is a magic board and another is terrible (for them).