Is Thunderbolt 4 with DisplayPort not compatible?

A MB Pro (14-inch, 2023, M2 CPU) has TB4 ports with DisplayPort protocols.
How do I connect it to a 24" LED Cinema Display (Model A1267), which was on sale from 2008 - 2010. The display has a MiniDisplay Port.

To use a Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter doesn’t work.

You’re overthinking it. Thunderbolt (whether version 1, 2, 3 or 4) will automatically become a DisplayPort video port when connected to a DisplayPort monitor (so-called “DisplayPort alternate move”). And mini DisplayPort is DisplayPort, just with a smaller connector.

The Cinema Display has one of Apple’s hard-wired cables featuring mini DisplayPort, USB and MagSafe power.

You won’t be able to use the MagSafe connector at all with a modern Mac, so you can ignore that.

The USB connector is type-A. You can use an adapter or connect it to a hub. But since it’s USB 2.0, you probably don’t need that either.

The remaining connector is mini DisplayPort, and is not Thunderbolt of any kind. So no Thunderbolt adapters. You want an adapter that goes from USB-C (or Thunderbolt-3/4) to mini DisplayPort. Your Mac will sense that the other end of the cable is not a Thunderbolt device and will switch the port into DisplayPort “alternate mode”.

A quick Amazon search finds many adapters including this one (no recommendations, just search results):

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You may want to check this conversation, Apple Cinema Display (2008) and Mac mini M2 - #8 by gilken, about connecting a Cinema display to an M2 Mac.

One bit of caution. When Apple was at TB and TB2 levels and MiniDisplayPort was popular, many adapters from China simply DID NOT WORK. I know you say your adapters are from Apple, but it’s important to make certain all your cables and adapters share the correct data- and power- transmitting capabilities, not just the same pins and shapes to their connectors.

For example, one time back in the TB2 or 3 era I had several successive failures attempting to migrate data from one laptop to its new replacement. The problem? Apple’s cable that LOOKED like a TB3 cable was not that at all, and not even “certified” as a data-transmitting USBc 3.2 cable, but rather just a power/charging cable!

Thank you @jsrnephdoc
I have now ordered this adapter USB C to Mini DisplayPort Adapter 4K60Hz - USB-C Display Adapters | United Kingdom

The manufacturer says it will work. I will report back here on Tuesday.

Thanks to @gilken for pointing me to the brightness issue that may come up. should help if needed.

Belatedly, it’s time to report back as promised:
Beforementioned adapter works just fine, no problems with screen brightness.