to make a "full normal install that is portable"
make your installer USB.
unplug or disable all the internal drives.
boot installer USB, plug in external drive.
install as you normally would.
it should be able to boot on most machines, assuming the open sourced drivers work.
Nvidia based video cards might be an issue depending on the card chipset.
if you install the Nvidia closed sourced drivers, then you might have issues on non Nvidia systems.
this is NOT the same as a live USB + a persistence save location.
a live USB is easier to make, and good for testing, but may not be as configurable as you want
to make a live USB, you can use the many tools listed at the pendrivelinux site, only some setup the live+persistence setup.
some of the tools can download and verify the iso files, and it is possible to make a multi distro live USB on a single large flash drive.
so your next decision will be, live or full? then pick a distro and go for it.
I think the OP is looking for a Live USB or portable OS (similar to Windows-to-Go), is that right u/FrostyHue ?
If so here is Fedoras documentation for doing so https://docs.fedoraproject.org/en-US/quick-docs/creating-and-using-a-live-installation-image/index.html
The gist is Fedora Media Writer can be used to create this. What you may want to research and focus on is how the Live image uses the space on the USB stick. I think it just uses the whole disk, partitioning what it needs for a few things and then leaving whats left for root or root and home partitions. If not you may need to find a way to determine the sizes so that you have enough storage space on the USB itself.
Yes that is correct!
I will try the methods and will see if everything is working properly.