Disambiguation of a sentence

I am working on a math problem where I need some help with the English to finish my solution.

I have an ambiguous sentence:

At least one student attended every lecture.

My attempt to disambiguate this sentence are the following two possible meanings for this sentence:

(1) There exists a student, who attended every lecture.

Here I am trying to make clear that there is at least one student who attended every possible lecture.

(2) For every lecture, there was a student present.

Here I am trying to make clear that for every lecture which was held, at least one student was in attendance.

Would these sentences be adequate for what I am trying to say? Do you have any better suggestions?