I can’t install MacOS on my external SSD.

Another option would be to boot into Internet Recovery Mode or by booting a macOS USB installer instead of running the installer while booted into macOS. I think the macOS installer can sometimes get confused when run from within macOS. Even when installing macOS from Internet Recovery Mode or a USB installer, I have found that many times the system will boot into the old macOS volume instead of booting into the new installation. In that case, you just need to access the Apple boot picker menu in order to select the partial macOS installation so that the install can be completed.

From your original post it doesn't seem like you erased the external drive while booted into Internet Recovery Mode which most likely would have resolved your problem. Here is the relevant section of the original post:

Howard Brazee wrote:

If I try a recovery install on the LaCie SSD, I get “The volume can not be used because it has an incomplete install on it. You must erase this volume with Disk Utility to continue.”

So I reboot and try running “Install macOS Monterey.app” again. 

Use macOS Recovery on a Mac with Apple silicon - Apple Support

How to create a bootable installer for macOS - Apple Support

You should use an app like Carbon Copy Cloner to make a bootable external drive. When you make the clone be sure to mark it to be bootable (called legacy bootable drive). Click on the destination drive and note the menu to make it bootable. Be sure it is formatted APFS, GUID partition map.

It was not mentioned in the original post which appeared to be quite explicit, so thanks for confirming. However, when erasing the external drive you need to make sure to erase the whole physical drive as GUID partition and APFS (top option). Within Disk Utility you will need to click "View" and select "Show All Devices" so that the physical drives appear on the left pane of Disk Utility.

Do not erase the whole internal physical drive on an M1 Mac since it contains special boot files to access the other boot options (such as Apple boot picker menu, recovery mode, etc.). Erasing the whole internal physical SSD on an M1 Mac will require access to another Mac in order to "Restore" the M1 firmware. It is Ok to erase the whole external physical drive.

It is possible that the Lacie external drive is just not compatible as a macOS boot drive. Apple computers tend to be very particular about the drives used for booting macOS. Or may there is a compatibility issue with an adapter, cable, dock/hub, etc. Just curious if you had been using the Lacie drive before this without any issues? I know that macOS 12.x Monterey does have some compatibility issues with some USB3 drives. There have also been some compatibility issues with M1 Macs and some external drives. Make sure to connect the drive directly to the Mac and disconnect all other external devices in case one of them is causing a problem. You may also want to try a different external drive. There is a chance the Lacie drive is bad which is most likely to be noticed during an OS install or when booting macOS from the drive.

Howard Brazee wrote:

I've been using that drive for data for some time.

Which may mean the drive is possibly worn out or failing.

It is formated each time with GUID partition and APFS. I see that it has "Container disk7" and "LaCie SSD volumes within LaCie Mobile SSD Media, and LaCie SSD & LaCie SSD - Data within LaCie SSD.

Unfortunately I have never had time to install macOS to an external drive on my M1 Mac, so I'm not certain if the layout is correct. I've never seen the "Lacie SSD volumes" item before, but this might just be how an external M1 boot drive appears. On the Intel Macs the "Lacie SSD" and "Lacie SSD - Data" volumes would be immediately after the "Container disk7".

It is plugged in directly, and I can use it as a hard drive right now, if I want.

Like I said Macs are very picky about the drives used for booting macOS. Just because the drive works as an external data drive does not mean it will work as a macOS boot drive. In theory it should not matter, but in reality that may not always be the case.

Of course you can always try the Carbon Copy Cloner option as a test, but make sure to select "Legacy" mode so the external clone drive is bootable.