Was my desktop activity equivalent to my server activity
Most of the time, yes, the desktop update is equal to
apt-get update && apt-get upgrade.
apt-get upgrade will not install packages like kernel upgrades that are optional but highly recommended, while Update Manager will. You have to use
apt-get dist-upgrade for that, in which case you would have a true equivalent.
If you use
apt-get upgrade and kernel upgrades, etc. are available, you will see a message like
The following packages have been held back:; that's your cue to run
apt-get dist-upgrade if you wish to upgrade those packages.
Note that a number of types of cloud servers do not use their own kernel, but their physical "host" machine's kernel (OpenVZ, vserver, LXC, Xen paravirtualized); other types behave like physical machines themselves (Xen HVM, KVM).