[libc] Remove integration test target from check libc.

Summary:
This is failing on the asan build because we use `-nostdlib`.
I also took this opportunity to make the target name match the naming structure we've been using.

Reviewers: sivachandra

Reviewed By: sivachandra

Subscribers: mgorny, tschuett, ecnelises, libc-commits

Tags: #libc-project

Differential Revision: https://reviews.llvm.org/D81029