However........ It would seem in my very limited knowledge of all things antennas that a directional antenna would give me the greatest amount of gain.......... Just how directional is a directional antenna?
A very precise direction with an unmeasurable degree of error?
A very close direction with a narrow angle of directional accuracy?
A wider more tolerable degree of adjustment?
A Yagi, is the most directional type of antenna in common use. Yagi type is the same type commonly used on chimneys for terrestrial TV reception and produce figures of around 20:1 gain. That's a 20 to 1 better signal strength when pointing directly at the transmitter, that when not pointing at the transmitter. The more elements to the antenna, the greater the gain figure, the narrower the reception angle.
In the early days of wifi, a relative living a little over half a mile away and almost LOS, needed Internet access, but with lots of other wifi in between. To provide that access, I built a pair of home-made 7 element Yagi antennas - 5x directors, 1x active, 1x reflector at the back. These plugged into adapted wifi routers. To limit coax losses, I co-located the routers in water tight boxes on the masts. Both radio amateurs and wifi is an amateur band, so permitted for us. It was mostly experimental, I had ideas to set up a national high speed packet data network at the time, but it worked very well indeed for several years.
Licence free, limits how directional and the gain of an antenna the public can use - it doesn't prevent suppliers making wild claims about the gains of their antennas. I think you are suggesting you have reasonable reception of local masts outside your home, that alone suggests that almost any antenna will work, if mounted outdoors, at a reasonably high spot - but keep in mind that losses in coax cables are large, connecting router to antenna, so the 'high place' needs to be not to far from your router. It might be possible to improve the directivity and gain of such an antenna slightly, by adding a suitably sized sheet of alloy to the back of such a ready made, as a reflector element.