You're both semi-right, semi-wrong. Winslow was brought in and the draft pick was dependent on him being on the opening day roster. Then it was obvious that he couldn't practice much because of his knee. He was asked to take a paycut, but was still on the final 53 when it was set (correct, or incorrect? There's been some discussion on this one for a while).
The team asked Winslow to take a paycut... he didn't. He wasn't cut immediately, because there weren't better options out there. Morrah was cut, so it was Miller, McCoy, and Winslow that were the final 3. Then Cleveland decided to drop Moore, and THEN Winslow was cut. It was sort of a multi-part situation.