Contest Math--at random vs uniformly

I recently worked on a math contest at a high school level (roughly AMC 12 caliber). In one of my questions, I used the phrase,

"...with the third vertex $P$ being chosen at random inside the square."

I was talking about the problem with a friend, and he said the question was underspecified, as I should have said uniformly.

I was under the impression that this was a reasonable assumption given the wording and necessity of a unique solution, but I could be wrong. I used the wording of a question from the 1989 Putnam exam, shown below:

Was my question well-posed, or is my friend correct?