Does that mean PingCAP can use the software including third-party contributions without having to follow the terms of the Apache license themselves?

I don't know what the intent of the CLA is but that's what it looks like to me.

Curiously, the copyright and patent grants in the CLA are almost identical to Apache 2.0:

2. Grant of Copyright License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable copyright license to reproduce, prepare Derivative Works of, publicly display, publicly perform, sublicense, and distribute the Work and such Derivative Works in Source or Object form.

The CLA though is not the Apache license. In particular the biggest section, the rules for redistribution is not in the CLA. And since the CLA states you retain your copyright, I don't see what the point of the CLA is except to give PingCAP a non-apache license to use the code how it wants.

Somebody correct me.