As a general rule, to change the layer's geometry without creating a new layer (apart from doing it manually), you can use Edit Features In-Place:
In the Processing Toolbox (Menu Processing -> Toolbox), activate the button Edit Features In-Place. You have then a list of algorithms that can modify the geometries of existing layers (not creating a new layer).
The tool you need is Split with lines
.
Be aware that not all algorithms can be used for that and be careful because this overwrites your data. There is a reason why by default, a new layer is created.
Another option might be modifying the layer's geometry with Field Calculator, but in this case, this would be a bit more complex.