Ran out of fuel and now it will not start

Turn the key from off to run a few times to get the pump to prime the entire system.

It has to push out the air and get to the injectors before it can start. Because it is not starting the pump is not running constantly. Basically the pump primes the fuel system then turns off. Then you crank the engine and the engine fires on the fuel in the lines for a second or so. As soon as the crank sensor sees the rpms rise faster than cranking rpms the computer says "hey the engines running, turn on the pump".
By turning the key on and triggering it to prime it should start. That is unless the car is at a strange angle so the gas cannot be drawn up by the pump. That takes a pretty steep angle though.

Any type of backfire suggests a timing issue. Hate to say it but you will probably need to have it taken to a shop and let them scan the system to see if something has failed. They can tell a lot about what is wrong in a few minutes where it would take you hours and taking parts of the engine apart.

The inertial cut off switch that your other half pressed can go bad but it's rare. Pressing the button simply resets it though so it's probably not the problem.

I talked to a couple folks who did suggest that it's fairly common for a couple of the wires in the engine harness where it runs down along the block to chafe through and fail.

Sorry I can't be much help. This vehicle doesn't show in the US databases we use so I've been getting information through a friend in Paris. The problem is he decided to go on "holiday". If I can find a reliable information source I'll try to help. I can say there are a couple Peugeot groups online that might offer more help than what I can at the moment.

Please let us know happens so it will help others.

Best, Ken