Definition of Projection Measure in the characterization of strong approximation Resistance in a paper by Khot et al
Section 1 of the paper, in which the first definition appears, is introductory. The formal development starts at Section 2. The part of the paper starting at Section 2 is completely self-contained. Therefore, if there is any mismatch between a definition in Section 1 and a definition elsewhere, the latter is the correct one.

In theoretical computer science, papers are published in conferences. Conference reviewers cannot be expected to read 50-page strong manuscripts, and so the argument has to be described in an elaborate introduction. This is why there is so much technical content in Section 1. The development there is simplified and more leisurely to facilitate reading. The authoritative part of the paper is Section 2 and forward.