/var/lib/mysql folder is missing
It's likely that you need to install the mysql-server package.
yum install mysql-server
You can search for packages by name with some of the following yum commands:
# list all packages starting with "mysql"
yum list mysql\*
# search package names/summaries/descriptions for "mysql"
yum search mysql
The packages which you did install likely provided only the client and client libraries, which you could use to connect to a mysql server located on another host.