Since you are having this confusion, I think it helps to consider the concepts of zero, infinity and "undefined".
In the most basic sense, division is the opposite of multiplication. Thus, the fact that 2 x 3 = 6 implies that 6 / 3 = 2.
1 x 0 = 0. Applying the above logic, 0 / 0 = 1. However, 2 x 0 = 0, so 0 / 0 must also be 2. In fact, it looks as though 0 / 0 could be any number! This obviously makes no sense - we say that 0 / 0 is "undefined" because there isn't really an answer.
Likewise, 1 / 0 is not really infinity. Infinity isn't actually a number, it's more of a concept. If you think about how division is often described in schools, say, number of sweets shared between number of people, you see the confusion. If I go around some people giving them 0 sweets each, how many people do I need to go around until I have given away my 1 sweet? An infinite number? Kind of, because I can keep going around infinitely. However, I never actually give away that sweet. This is why people say that 1 / 0 "tends to" infinity - we can't really use infinity as a number, we can only imagine what we are getting closer to as we move in the direction of infinity. However, in this case, the number of sweets I have is never changing, so I'm not really getting closer to anywhere. Even this logic doesn't really work.
The long and short of it is that 1 / 0 doesn't really make sense as a calculation. When we do use the notion of infinity we tend to use positive infinity where it doesn't matter purely by convention. However, if you think about it too hard you start to get into philosophy and stuff, like "what actually is infinity?" and "wait, what is a number"?
The things people are talking about where it does are different ways of using numbers so they don't really count. For example, in the trivial ring, there is only one number, which works like a 0 (add it to anything and you get that thing) and a 1 (multiply it by anything and you get the same thing again) and makes sense because you can only add it to or multiply it by itself to get itself. It's pretty boring actually, but in that case this one number - let's call it x - is both 0 and 1, so 1 / 0 = x / x = x because everything equals x. As you can see, this is a bit of a cheat because we don't even have enough numbers to have a notion of 1 / 0 in the way you're thinking of it.