Well, I believe the authors do their best to convey their ideas, but due to the fact that the notation is mostly informal it's easy to miss details. Also, different institutes might use their own special notation to describe the same ideas and even individual authors prefer some notation over another.

All in all, imprecise probably was the wrong term. The statements are precise for the people working in this specific "bubble" of research, because they might know the implicit assumptions made and in the worst case, the only one who knows the implicit assumptions is the author.