First approach is more common, hence the genre name "Scrolling Shooter" - the player remains static (on one axis), while the stage is sliding past them.
In older games this also implied that stage elements (enemies, power-ups) were also loaded just as they were to pop into the view, and destroyed as soon as they are left behind the player.
This also helped to keep memory usage low and logic simpler, since it was known that any coordinate would always be in fixed range slightly larger than screen.
Second approach is less common for shooters but occurs more often in other genres (e.g. platformers) - if your player is moving through stage, it totally makes sense to just scroll around the camera alongside with them.
Functionally... nowadays these should not differ much.
A response from developer would help more, but generally engine is going to render images that overlap the view area one way or other.
With numeric values being stored in double-precision floating point format nowadays, you are also not going to see any kind of number precision problems unless your coordinates exceed about... 10�6. Which is an extremely unlikely scenario.
To sum it up, both will work, and in your game you choose, what feels right. Moreover that methods can be swapped quite easily.