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SKYLANE
I selected this Tripp Lite cable to save some money over buying another Apple cable & adapter.
You have a USB-C to Mini Displayport cable. It is just a dumb display cable. It doesn't carry a Thunderbolt signal or any type of signal other than video for a display.
Thunderbolt 2 on the Mac uses a Mini Displayport form-factor and Thunderbolt cables plug into the Mini Displayport port. But the port has multiple functions. It can be used to connect Thunderbolt devices, and it can be used to connect BOTH Thunderbolt devices and a display (daisy-chained) and it can be used without Thunderbolt at all just to connect a display.
To use Thunderbolt on a Mac with Mini Displayport ports requires cables/adapters that support Thunderbolt in that form-factor. Your Tripp Lite cable does not support Thunderbolt. It's just a dumb display cable. It won't pass a Thunderbolt signal.
Thunderbolt on newer Macs -- and on some devices like a Thunderbolt-3 LaCie drive -- passes its signal over a USB-C connection instead of Mini Displayport. As with the Mini Displayport port on your old Mac, the USB-C form-factor is used, but you need a USB-C
Thunderbolt cable in order to use Thunderbolt on a new Mac. If you use a dumb USB-C display cable on a new Mac, you can only use it to connect a display. If you use a USB-C data-cable on a new Mac, you can only use it to connect USB devices.
Your Tripp Lite cable was designed to work as a display-only cable for USB-C ports. It passes a display signal from a USB-C Mac or PC to a display that has a Mini Displayport input. It does not pass a Thunderbolt signal. It would not work to pass any other sort of signal than video.
It only works in one direction. That is, it won't even work to connect a Mini Displayport Mac to a USB-C display. It only works to connect a USB-C Mac or PC to pass a video signal to a display.
So, your original solution, connecting a Thunderbolt cable to your old MacBook Air and then a Thunderbolt/USB-C adapter to connect to the LaCie drive is the right one.