It looks like your MacBook Air connects fine with your Apple Thunderbolt Display, but your'e unable to see video when connected to another TV or projector. I'm happy to provide some help for this issue.
I recommend following the article below, specifically the "Blank or unusable display" section:
Get help with video issues on external displays connected to your Mac - Apple Support
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Try these steps if you don't see an image on your display. These steps can also help if the image on your screen repeatedly turns on and off (flickers), if horizontal lines appear (snow), or if the image is distorted (torn or scrambled).
These steps likely won't help if the issue is only happening in one app. If you see similar issues in only one window or app, check with the app's developer for updates or more help.
Check the connections to your Mac and external displays:
- If you're using an Apple notebook, try connecting its power adapter.
- Make sure the external display's power cable is securely connected and that your display is turned on.
- If you're using a Mac Pro (Late 2013) make sure your displays are connected to the right ports.
- If you're using a display hub, switchbox, or "KVM," try connecting your display's video cable directly to your Mac instead.
- Disconnect the video cable where it plugs into your Mac, then plug it back in to reseat the connection.
- If you're using a video adapter, unplug the video adapter from your Mac, then plug it back in to reseat the connection.
- If you're using more than one video adapter to connect your display (the adapters are "chained" together), try connecting the display using only one adapter if possible. Some video adapters can't be connected together. For example, a mini DisplayPort to DVI adapter can't be connected to a DVI to HDMI adapter.
- If your display has more than one video connection, see if using another connection on the display works. If possible, check to see if using a different display or a different adapter works.
- Try using a different cable that you know is in working order. Check with the display's manufacturer to make sure you're using the cable they recommend.
- Restart the Mac with the display connected.
If you're using an external display, sleep and wake your Mac to make it check for connected displays:
- Press the power button on your computer to put it to sleep, or choose Apple menu > Sleep.
- Wait a few moments, then press a key on your keyboard or click your mouse or trackpad to wake your Mac.
If you're using more than one display, and you can see the image on one of the displays, you can also use Displays preferences to check for connected displays.
If there's no image on your display, try adjusting your display's brightness or contrast. If there's still no image, or the image appears torn or scrambled, try selecting a different video resolution in System Preferences.
- If you're using an Apple Display, press the increase brightness key on your Apple keyboard. To adjust the brightness of a secondary display, press Control-F2.
- If you're using a display that's not made by Apple, check the documentation that came with your display to see if it includes built-in controls for adjusting its brightness or contrast.
Change your display resolution
- From the Apple menu, choose System Preferences.
- Click the Displays icon.
- Click Default for display. If you see more than one resolution listed, choose a resolution and refresh rate. If the display turns off when you change resolutions, press the escape key to undo the change.
If you can't change the resolution of your display because you can't see an image, restart your Mac in safe mode to reset the display resolution to defaults.
If starting in safe mode doesn't resolve the issue, reset your Mac's NVRAM and SMC to reset the video ports on your Mac to their defaults.
If your display works only when your Mac is started in safe mode, and you have display-related software installed, check with the software's developer for updates, or try temporarily uninstalling it.
If you've tried these steps and your display still isn't working, contact Apple Support to get more help.