You may have unexpected problems if you do not use the uuids. Like in your case, the live gives them different names, this may mean that if you are trying to chroot your system from a live and use your fstab, the disks will be wrong. Also if you are dual booting and the disks swap names between the distros, I'd be worried that an upgrade of one of them may break it. What has happened to me is that an upgrade of the kernel and/or init system may rearrange the names of devices. This because there probably is some race-condition that changes because e.g. systemd changes. So what would happen if one distro upgrades its systemd to the same version as the other, will the names of the devices change? Maybe.

I'd do the install as you'd normally do. Then if you look at any of the installs fstabs you should only see uuid lines in there, if you don't, change them to uuid lines. If you do something manually outside of the fstab, do not reference the disk device names, only the uuids.