Malcolm, thank you so much for your response. However, I have the exact USB-C to Mini DP cable that you linked to here and I have a proprietary Apple connector like the one you show here - except that it also has an HDMI and a USB-3 slot, in addition to the USB-C slot that I plugged the Cinema USB 2 cable into.
My LED Cinema specs say it has a: Mini DP with audio, Magsafe (not used) and a USB 2.0.
My MacAir M1 (Technical specs on Apple's web site) says it has: "Two Thunderbolt / USB 4 ports" with support for: "Charging, Displayport, Thunderbolt 3 (up to 40 GB/s) and USB 3.1 (Gen 2 (up to 10 Gb/s). All way too confusing because both of these "USB 4 'Thunderbolt' ports appear to be USB 3.1 ports -- which is what the "System Report" says is on this machine when I click on my Apple icon.
May I ask - Do you actually have an MacAir M1 connected to an LED Cinema Display or is your logic saying your suggestions above should work fine?
What I really need, I think, is someone who actually has their MacAir M1 (2020) connected to a LED Cinema Display.
Unless I'm missing something, I am about to give up on this.
Thanks again...willing to still listen to how to make this happen.
Jim