Within the 3 different lengths of centerfire actions - short, long, and magnum - there are 3 "normal" bolt face diameters - "223rem" .384", "standard" .473", and "magnum" .532". There are longer actions for specialty cartridges as well as larger bolt faces within these lengths.
Not all of these combinations cross over, as an example, I don't believe there are any long or magnum actions with .223rem bolt faces, and I don't recall anything but magnum bolt faces in magnum length actions (sure seems like there are, but I don't recall any right now).
So yes, you'll need to get the right bolt head for your cartridge. Luckily, you have a Savage, the bolt heads are readily available for purchase.