Full Install USB
The official versions that you point to are old and result in a Live non Persistent install. Nowadays Rufus offers Persistent partition installs for Ubuntu 19.10 and later. These are not limited to 4GB.
If you want to run Ubuntu from USB just like from an internal drive you can make a Full install to USB.
It is safest to first unplug your internal drive and then insert your USB. You can then install as you would to HDD, either automated install or using "something else".
Advantages of a persistent install:
1) You can use the persistent pendrive to install Ubuntu to another computer.
2) A persistent install takes up less space on the pendrive.
3) You can reset the pendrive by overwriting the old casper-rw file with a new one.
4) The install to pendrive takes less time.
Advantages of a Full install:
1) You can update files and upgrade Ubuntu versions.
2) If you have problems or wish to modify, the solution is the same as with an internal install, (You can ask for help in these forums).
3) No ugly startup / install screen.
4) Better security, you can use full encryption.
5) You can use proprietary drivers.
6) Hibernation works.
7) A persistent install is often limited to a 4GB casper-rw and a 4GB home-rw persistence file, to get more persistence requires persistence partitions. Once casper-rw is full, the drive will not boot.
8) Faster boot.
9) You can run VBox and use virtual machines.
Note that once booted, both methods run at about the same speed.
Full Install Methods
Two methods for creating Full install pendrives: How to make a live Ubuntu 18.04 USB with a persistent storage of more than 4GB and How to fully install Ubuntu on USB Flashdrive?