How do I keep Centos at version 6.3?

How do I prevent automatic upgrades from occurring to versions of the server OS when performing yum update?

The upgrade from CentOS 6.3 to CentOS 6.4 is a point release. As the link says

Old point releases are never supported. If you want/need to "freeze" at an old point release you are on your own.

What this means is that if you want to stick at C6.3 then no updates will be provided so what you might as well do is disable all the yum repositories. You can do this by adding enabled=0 (or changing the enabled=1 to 0) to each of the [...] sections of the files in /etc/yum.repos.d.

This is a really bad idea. You're on your own. You don't get bug fixes, backported security fixes and updates etc.; you lose your herd immunity.

Is there a "standard" location on the internet for finding what changes occur in a version upgrade?

You can subscribe to the CentOS Announce mailing list which will provide you with a list of updates. The answer to this question is relevant here too. You can check the Centos release notes pages.

Am I being paranoid about server version changes with respect to stability?

No, like all software systems operating systems have their bugs and vulnerabilities. Just like any other piece of software you should evaluate and test proposed changes in your environment before deploying them to your production systems.

You need to evaluate the risks of 'fixing' your OS version and receiving no updates (which over time leaves you open to increased numbers of vulnerabilities) against the risks of updating via a testing process to ensure that nothing breaks when an update is applied.

yum --releasever=6.3 update

will stick to 6.3 repo.

For CentOS 7.x, the yum --releasever=X update method does not seem to (always) work. This is what I had to do instead:

Stick to version

1. Check http://vault.centos.org/ for available versions (eg: 7.3.1611)
2. Edit /etc/yum.repos.d/CentOS-Vault.repo so that it includes a full set of repo definitions for the chosen version (7.3.1611 in our case). You should have 5 repos named [C7.3.1611-X].
3. Disable all CentOS repos: yum-config-manager -v --disable CentOS\*
4. Enable the 7.3 repos you added: yum-config-manager --enable C7.3\*
5. yum update

Obviously this will prevent any security updates.

Unstick

1. yum-config-manager --disable C7.3*
2. yum-config-manager --enable base

(YMMV with these instructions depending on what was disabled above)