You're right to think that the open string content of a theory is essentially a choice. The strings are the fundamental constituents, and different theories have different building blocks, by definition. But there are some constraints on what you're allowed to have, as I'll now explain.
Firstly, I think that website has things a little back-to-front! Here's what it's trying to say. If you have open strings in a theory, you must have closed strings, because a loop of open strings looks exactly like a closed string. But the converse is not true!
In particular you shouldn't really think of the closed string "snapping". In string theory, the tension is usually taken to be a constant. This means that the string doesn't stretch and snap like a normal string. Instead it has some quantum probability of splitting into two open strings, if open strings are allowed in your theory.
So when are open strings are allowed? Originally people thought that you could only get open strings whose ends were allowed to roam over all of spacetime! This is because any constraint on the ends of the string breaks Lorentz invariance. For historical reasons the type of string theory including these open strings is called Type I.
Towards the end of the 1980s people realised that open strings could also end on dynamical $D$-branes, thus preserving Lorentz invariance. This meant that you could have open strings in theories where previously they didn't crop up. In particular Type II string theories contain open strings which end on types of $D$-brane. For technical reasons to do with the field content of the theory, only certain dimensions of $D$-brane are allowed!
But even with $D$-branes there are still some string theories in which you can't have open strings at all! These are the heterotic theories. In such string theories, the left and right moving oscillations along the string have different amounts of supersymmetry. It turns out this is fine if the string is closed, but not allowed for open strings! So there are no open string states in a heterotic theory.