A Distributor would generally take a single flow source and distribute it to different cylinders..
..like a spark distributor takes an electrical signal and distributes it
to many plugs.. [ through the cap/rotor].
..Same deal with a fuel distributor, but you now talking fuel vs. spark..
A Manifold also is used to do the same , but is more of a Collector vs a Distributor in the sense that it does not involve any Timing configurations in its workings..

On the exhaust, we have the opposite of distributor in the sense
that you are now taking a flow from many sources [each cyl., in this case] and bringing then into one artery [ [the exhaust]
So, this is a Collector vs. Distributor..
On small Aircraft engines , they are actually called exhaust Collectors...vs, exhaust manifolrds..
Anyway , something of interest...