Great!
It sounds like you have things on track.
The reason I suggested that you check the swap status with swapon -s is that Ubuntu 18.04 can install with multiple swap options.
If you assign a dedicated swap partition, or install with LVM/LUKS encryption, then 18.04 will use a separate swap partition, but since version 17 Ubuntu can also use a simple swap file.
From the swapon -s response that you posted, it looks like your MATE 18.04 installation is using a simple 2 Gig swap file (2,097,148 1K blocks).
There are several ways to enable swap using a swap file in Debian based releases, but I suspect that they handle it like they did in the earlier Ubuntu 17 release using a simple /etc/fstab entry.
If you want to shift back to using the traditional disk partition method, look in your /etc/fstab for an entry like:
/swapfile none swap sw 0 0
... and change it to something like:
UUID=xxxxxxxx-xxxx-xxxx-xxxxxxxxxxxx none swap sw 0 0
... where the UUID matches your new swap partition (just like you did with 16.04)