The 27" Apple Cinema Thunderbolt Display (A1407)
https://manuals.info.apple.com/MANUALS/1000/MA1594/en_US/thunderbolt_display_getting_started.pdf
Resolution: 2560 x 1440 @ 60Hz
Ports: 3 USB 2.0, 1 Thunderbolt 2, 1 FireWire 800, 1 Gigabit Ethernet on the back of the display
Camera and Microphone: Built-in FaceTime HD camera (720p) with microphone
Speakers: Built-in 2.1 channel speakers with 49 watt output
Connections: MagSafe and Thunderbolt 2.
MagSafe is used to power MacBooks (if port is exists) while connected to the display.
Thunderbolt 2 provides video and audio signal and port connectivity to the back of the display.
To connect this display to a M1 MacBook Air or M1 MacBook Pro requires a Thunderbolt 2 to USB-C adapter for the video and audio and port connectivity to the back of the display. The MagSafe connector would remain unconnected since no MagSafe port is available on the device.
To connect this display to a M1 Mac mini requires a Thunderbolt 2 to USB-C adapter for the video and audio and port connectivity to the back of the display. The MagSafe connector would remain unconnected since MagSafe is only found on some MacBook computers.
The 27" Apple Cinema LED Display (A1316)
https://support.apple.com/kb/SP597?locale=en_US
Resolution: 2560 x 1440 @ 60Hz
Ports: 3 USB 2.0 on the back of the display
Camera and Microphone: Built-in iSight (640x480 VGA) with microphone
Speakers: Built-in 2.1 channel speakers with 49 watt output
Connections: MagSafe, Mini DisplayPort, and USB.
MagSafe is used to power MacBooks (if port is exists) while connected to the display.
Mini DisplayPort provides video and audio signal.
USB provides connectivity to the USB ports on the back of the display.
To connect this display to a M1 MacBook Air or M1 MacBook Pro requires both a Mini DisplayPort to USB-C adapter for the video and audio and a USB to USB-C adapter for the USB ports connectivity (since there are no USB ports on these devices). The MagSafe connector would remain unconnected since no MagSafe port is available on the device.
To connect this display to a M1 Mac mini requires a Mini DisplayPort to USB-C adapter for the video and audio. You would connect the USB directly into one of the USB-A ports on the mini to enable port connectivity to the back of the display. The MagSafe connector would remain unconnected since MagSafe is only found on some MacBook computers.
Please note that because the Thunderbolt Display can only use a Thunderbolt 2 to USB-C connection and the LED Display can only use a Mini DisplayPort to USB-C connection. The quickest way to identify which display you have is to look at the back of the display itself. If you see an ethernet port, it's the Thunderbolt Display. If you don't see the ethernet port, it's the LED Display. I note this because the connector for Thunderbolt 2 looks very similar to a Mini DisplayPort connector. They are not the same.