First things first: any EM radiation will attenuate as it leaves an antenna. It will follow the inverse-square law for a start and also the real world is not quite the ideal 'free space' environment that is generally used (but is close enough for practical purposes). This is so regardless of the initial power and direction that the radiation is launched from the antenna.
Where the direction and power come in is that, if you start off with a higher power signal in one direction then the attenuation will still be the same but at some given distance away in the given direction, because you started with a higher power, you will receive a higher power.
When you design an antenna, you need to think about where you want the signal to go. An ideal "isotropic" radiator will radiate equally in all directions and this is usually used as the "gold standard" for measuring the gain of any other antenna design. A Yagi for example will not transmit very much power directly out of the ends of the active element but will send more power at right-angles to the element. If you stand the element vertically, you can imagine that this will send in a doughnut shape out horizontally in all directions with up and down representing the "hole" in the doughnut.
Often the way an antenna radiates is shown in radiation plots that show the vertical and horizontal radiation patterns with the relative strength shown as a circle around the centre. The isotropic radiator will show up as a circle in both the vertical and horizontal plots. The Yagi I mentioned above that is oriented vertically will show as a circle in the horizontal plane and a couple of "wings" in the vertical plane.
These 'wings' are generally referred to as 'lobes' and the point of antenna design is to make the radiation patterns suit your requirements. You mention a satellite antenna - there you want a single lobe (or beam) that points in one direction and it as narrow as needed (if the satellite is geosynchronous then the lobe can be very narrow; if the satellite moves such as the LEO satellites, you need to move the antenna to track the movement and so you might make the beam a little wider to lower the requirements for accurate mechanical tracking mechanisms).
When you talk about the gain of an antenna, you generally mean the relative strength of the main lobe to the isotropic radiator.
Therefore your statement in the 2nd paragraph is both right and wrong. It is right in that the gain means the main lobe is better (or worse) than the isotropic radiator and as it is stronger it can travel "further" before the attenuation reduces the signal level to below where it can be detected. However you are wrong when you say it has less attenuation - as I outlined above, the attenuation is always the same.
This also should sow that your 1st paragraph is also wrong in that the gain is measured along the main lobe which IS (generally) in a "certain direction".
Susan