A1466 matches a number of 13-inch MacBook Airs with model years ranging from Mid 2012 to 2017. All have old-style Thunderbolt 1 or 2 ports with Mini DisplayPort connectors that can act either as Thunderbolt ports or as Mini DisplayPorts, depending on what you plug into them.
A1407 would be an Apple Thunderbolt Display (27-inch). It requires Thunderbolt input, and it has a two-headed hydra-cable with an old-style Thunderbolt head and an old-style MagSafe head.
Normally you would plug the Thunderbolt head of the hydra cable into the Thunderbolt port on the MacBook Air.
If that cable is bad or flaky, you could try getting a Thunderbolt 1/2 cable and running it from the display's daisy-chaining port to the Thunderbolt port on the MacBook Air. A Thunderbolt 1/2 cable would have Mini DisplayPort connectors on both ends, but would be a higher-grade cable rated to carry Thunderbolt traffic.