Is coasting down hills in neutral actually saving me fuel or am I just being daft?

Your mate is right for fuel injected bikes. When the throttle's shut and the bike's in gear, the ECU just stops firing the injectors on the overrun, so engine braking costs literally nothing. Coast in neutral and the engine needs idle fuel to keep ticking over, plus you lose the free deceleration, so going down the same hill in gear is actually cheaper. That's why it feels backwards when you watch the light. Neutral steals the free braking, so you're idling longer and burning a trickle the whole way down instead of zero.

On a carb bike like yours it's a bit diferent, because a carb can't shut the fuel right off either way. Closed throttle in gear or idling in neutral, you're basicly on the idle jet either time, dipping a few pennies worth over the length of the hill. Honestly the difference is so small on a 125 that it won't decide wheather you make the station or not. The bigger deal for me is the tap. If you're down to half a litre you're almost certainly sucking from reserve alredy, so flip it to reserve deliberately rather than finding out mid-hill when the engine coughs and dies. And if the engine does cut out while you're rolling down that hill, you want to be in gear so you can bump start it with a bump of the clutch rather than coasting along dead in neutral praying the corner goes your way. Also worth thinking about what happens when you hit the bottom at neutral-coast speed with only your brakes doing the slowing on a long descent. Brake fade on a little bike is no joke on a proper hill. I'd leave it in gear, top gear, closed throttle, let the engine hold you back a bit, and put the difference down to paranoia tax. If you're that close to empty, the real fix is a five litre can in the top box til the tap situation gets sorted.

Not convinced the difference is 'a few pennies' on a carb 125 though. I had a CG125 for years and tested this exact thing out of boredom on my commute. Idling in neutral down that long drag near me consistently cost me more range than leaving it in top with the throttle shut. Half a litre in a tank that size is a proper chunk of your total, so the trickle matters more than it would on a big bike. Maybe the maths changes on a fuel injected machine, I wouldn't know, but on carbs I'd trust the fuel light over the theory.

Worked as a bike courier for two summers on a little 125 and learned its real range the hard way. What I did after running dry once was take a bottle with me on a quiet Sunday and deliberately ride it dry near home, noting the trip meter. Turns out my 'empty' was actualy another 40 km left, the light came on stupid early because of the tank shape. Once you know the true number you stop sweating every hill. Worth doing on yours before you trust the tap or the light again.

Update: tried leaving it in gear with the throttle shut like the CG125 owner suggested, and the tank went noticeably further than when I was coasting in neutral. Mate was right that the injection cuts fuel on the overrun, mine's a carb but it still seemed to sip less that way. Also going to do the run-it-dry test near home before trusting that light again.