How to check if kernel is properly updated/patched?

I have received an email from Digital Ocean to update the OS to mitigate the industry-wide security vulnerabilities known as Meltdown and Spectre

Quote from email: To simplify the act of patching, we have recently updated Droplets to utilize a GrubLoader. On certain legacy Droplets, this may cause issues if the kernel is not upgraded.

This is the link that I am referred to about the upgrade: https://www.digitalocean.com/community/tutorials/how-to-protect-your-server-against-the-meltdown-and-spectre-vulnerabilities

I am running CentOS 6, I have executed the following

yum update
yum update kernel

When I do uname -r I still get:

2.6.32-504.12.2.el6.x86_64

But when I do yum install kernel I get:

Package kernel-2.6.32-696.20.1.el6.x86_64 already installed and latest version

Does that mean the kernel is not updated accordingly?