You are absolutely correct that the container should be wider than it is deep. This is true for all cacti, and IMO, for all succulents. The weight of the thick fleshy stems and leaves can make a succulent top heavy, so giving it a broad base for support will help stabilize it. In addition, the potting mix in a wide shallow container dries out faster than that in a deep container, so it helps prevent root rot.
Root rot is what kills most succulents, so you are wise to be careful of how much water you give the plant. Another thing that helps is using a container made of unglazed clay, because the clay will absorb some of the water away from the roots, and it also allows air to get into the soil. Air is necessary for healthy roots.
Aloes are desert cacti, as opposed to jungle cacti. It doesn't rain often in the desert, but when it does, it's usually a real cloudburst. That's the environment your plant has evolved for, so when you water it, water it very thoroughly. Then, allow plenty of time for all the excess water to drain out, before you return the pot to its saucer. After that, it won't need water for a few weeks, maybe a month. In the winter, when the days are shorter, your plant will stop active growth. Water it even less during that time, because the roots won't absorb the water from the soil as quickly as they do while the plant is growing.
Being a desert plant, aloes also like a lot of light. You might even want to get a grow light to supplement whatever natural light you can give it.
After all of that, though, I must say your mom really has done a fine job with that plant. You might just want to ask her to tell you how she treated it, then try to replicate that as best you can.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams