Shared time machine drive on external disk.

I"m retiring my old Time Capsule. The most logical change would be to set up Time Machine on a network drive but it seems as though that doesn't work well according to Apple Support. They suggested I use an external drive. I purchased a drive with external power source (feature suggested by support that makes good sense). I set up Time Machine with it as the target. All worked fine until I tried to connect the MacBook to work on the shared drive. Now I have the following question:

Has anyone had any success in using an external disk on a Mac with a MacBook for backups? In Apple document Backup disks you can use with Time Machine - Apple Support the process for "Mac shared as a Time Machine backup destination" is explained. I don't seem to be able to get it to work. I keep getting a permissions problem indicated.

I did find another set of instructions from someone else on an earlier version of the Os that refers to making a folder for the shared backup. I thought that might be the solution but the external drive has limited me to read only. I suppose since it is being used by Time Machine? The article reference is: https://nandakusumadi.com/how-to-setup-shared-time-machine-drive/

I really didn't want to erase the drive and start over but can do so if the is a good chance of things working correctly afterwards.

Any successes or ideas?