Turning off the engine at red lights does save fuel. But it also does wear out the starter motor much faster, your starter wasn’t designed for this type of service. Leaving the key in the on position in a modern car while waiting for a green light doesn’t do any harm, but it doesn’t help either, the car will not start any faster.
If you have a manual transmission, then you can do this without holding up traffic, but if you have an automatic, then you probably are, unless you are watching the lights for the cross traffic and restarting when they turn yellow.
This was tested in Japan where some prototype cars had an automatic starting starter installed. when the car came to a stop, the engine would shut down. When the gas was pressed, the starter would engage and start the engine while still in gear (automatic transmission) and the car would go. This was done back in the 70’s with taxi cabs which meant carbureted cars. It didn’t work well at that time. It saved fuel but reliability suffered.
Some hybrid cars use this technique, but their starter motors are also the drive motors so they are very robust. starting the engine is no strain for them.
I could see doing this in places that typically have very long light cycles. California for example holds the red lights for what seems like at least 2 minutes. Around here, a red light typically will last no more than 45 seconds and could be as little as 15 seconds so shutting down would be a waste of time and effort.
If you really want to save gas, try matching your speed to the light cycles so that you hit every light green, that way you don’t have to stop at all. I do this and I get really good gas mileage around town, about on par with my highway mileage. My brakes and clutch last a really long time too.