Planning a build in the Lian Li TU150 and ideally I would have one 120mm front intake, one 120mm exhaust, and 2 x 120mm bottom intakes. The motherboard I'm using is the ASUS ROG Strix B550-I which has just one chassis header. I'm a little confused on how fan headers work - does this mean I can only have 1 case fan?
It might be pushing it a little, you have to check the motherboard manual to see what maximum current rating the fan header is and then look at the fans you plan to use and how much they are rated for.
Its also possible the AIO_PUMP fan connector might be switchable in the BIOS to act as a second normal chassis fan header, the VRM_HS_FAN header probably also works as a normal chassis fan header. The names are most likely just for easier identification, functionally they're probably identical but again you'd have to look in the BIOS to see what options they have for fan control.
Alternatively use a fan controller, although I haven't seen one that uses motherboard speed controls (though theoretically its perfectly possible) so that would require a fixed speed or software to control the speed.