The translation 'intention' is not that bad, but maybe, 'have in mind' will work more generally.
I have in mind to take a full rest during the summer vacation.
つもり is one way of making your statement accurate, and in many cases polite. In linguistic terms, this is called a hedge. In this particular case, everyone might think herself/himself is right, but they can actually never be sure about their knowledge. If you simply say
I understand that ...,
then it might later, for some reason, turn out that you actually haven't understood it. In that case, you would be lying. But if, instead, you have said
I think I understand that ...,
then your statement remains true. The latter statement is more accurate. And it is also polite in not saying a lie, and for being careful about your words. It is the same thing for Japanese sentences
分かっている 'I know that'
分かっているつもりだ 'What I have in mind is I know that' or 'I think I know that'