How do I install macOS Tahoe on an external drive when recovery mode shows macOS Sequoia?

I have an M1 MacBook Air, and I updated it to MacOS Tahoe recently. The internal drive is a tiny 115.44 gigabytes, and so I want to install MacOS on a 500 gigabyte external USB-C SSD that I have (Seagate OneTouch). When I go to recovery mode, it says "reinstall MacOS Sequoia" instead of Tahoe. When I select that option, it looks like it's installing on the drive until the end. After it finishes installing it restarts twice and then says the startup disk cannot be verified. If I try to manually set it to be my startup disk, it says that MacOS needs to be reinstalled on that disk. I've done this twice with the same results. I also can't reinstall Tahoe on my internal drive (which may fix the recovery having the wrong MacOS version issue) because the only way to do so is through the corrupted recovery mode, which says I cannot downgrade my internal disk. Another important detail: I live at my university, and their internet requires me to sign in with credentials, but the captive doesn't show up in recovery mode, not allowing me to connect to the internet. Instead, I plugged in an ethernet adapter connected to my dorm, and that allowed me to connect to the internet and at least allow the Mac to access Apple's servers (even though the installation was still corrupted). How do I get out of this situation? I've installed MacOS without an issue on this exact same external drive before and don't understand why there is a version mismatch? Thanks for any help you guys can give.

[Re-Titled by Moderator]

Original Title: Cannot install to external drive; recovery mode tries to install Sequoia when internal SSD is on Tahoe.