144hz display option not available from mac to monitor

You might want to take things you "read online" with a grain of salt.

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MacBook Air (M2, 2022) - Tech Specs - Apple Support

MacBook Air (15-inch, M2, 2023) - Tech Specs - Apple Support

"Simultaneously supports full native resolution on the built-in display at 1 billion colors and:

• One external display with up to 6K resolution at 60Hz"

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Use an external display with your MacBook Air - Apple Support

If you go back to the "2023 – macOS Sonoma 14" version of this article, it says "Use a USB-C port to connect one external display with up to 6K resolution at 60 Hz." The M2 MacBook Airs came out in 2022 (13' models) and 2023 (15" models), and this older version of the Support article explicitly refers to M1 and M2 MacBook Airs.

All newer versions of the article that mention higher refresh rates refer to MacBook Airs that have M3 or M4 chips – and appear to be referring to the capabilities of MacBook Airs with those chips. Chips your Mac doesn't have.

ceejay5303 wrote:

they advertise being able to output upto 4k 240hz

Apple never advertised that M2 MacBook Airs could support 4K monitors running at a 240 Hz refresh rate.

That's a feature that Apple advertises in the Technical Specifications for the M4-series MacBook Pros, and that Use an external display with your MacBook Air - Apple Support describes for M4 MacBook Airs. Even on those newer and higher-end machines, attaching an external display that runs at 240 Hz requires you to sacrifice; the cost of supporting that refresh rate is to reduce the maximum supported number of external displays.

ceejay5303 wrote:

i meant the cable advertisement, not the machine itself being able to output 240hz

A cable cannot increase the maximum refresh rate that a computer supports.

If a cable is built to support 4K @ 240 Hz, then it should be well enough constructed to support the 4K @ 60 Hz that the M2 MacBook Air will actually try to send over it. But that's it. When a cable vendor advertises that their cable supports 4K @ 240 Hz, they're saying that it can support such signals if your computer generates them … they are not promising that their cable will make your computer able to generate them.

EasyRes UEFI, DisplayConfigX, or SwitchResX offer additional resolutions.