When I use this library and try to simulate Mexico City in my browser it displays CDT where I expect to see CST.

Please show us that code so we can reproduce it. For example, you might be displaying an old date where CDT was in use.

Also, for issues with Moment Timezone specifically, it's usually best to raise an issue in the project's bug tracker first: https://github.com/moment/moment-timezone/issues

If there's a problem with outdated or incorrect data, it's likely to be in the library itself rather than the underlying tzdb. (I say that as the current maintainer of the library.)