Your DBA and you are both right and both wrong.

SQL hotfixes are cluster-aware, just like a SQL service pack.  The installer
will update binaries on all nodes and will update the database on the host
node.  You must start the hotfix installer from the node that currently owns
the instance you wish to update. SQL uses local binaries on each node, but
the installer knows this and will handle things appropriately across the
cluster.  The only time you run a local binary only update is when
rebuilding a failed node.  The Service Pack and Hotfix README files explain
how and when to do this.

That leads us where you are both wrong.  SQL hotfixes are applied to each
instance independently.  During the installation process,the installer takes
the instance offline.  You may have to reboot the cluster host nodes after
installation, but the installer will alert you if that is required.  Best
practices suggest a rolling reboot of all nodes prior to any SP, hotfix, or
OS patch to ensure a clean system.

If you have multiple instances, you will need to apply the hotfix to each
instance independently.  Note that Microsoft does support running different
instances at different patch and even version levels on the same cluster, so
you don't have to update all the instances at once.