Origin of the MIDDLEBROOK name

There appear to be two possible origins of the name as far as I can tell.

A topographical name derived from 'a dweller by the middle stream'.  It comes from the Old English middel for 'middle' and broc for 'brook' or 'stream'.

Bearing in mind the geography of Yorkshire and the adjacent counties it is highly likely that this is the origin of the bulk of the MIDDLEBROOKs (see Distribution of MIDDLEBROOKs).

From the Subsidy Rolls for Kent in 1334 is mention of Thomas de MIDDELBROKE.  Could this the source for the MIDDLEBROOKS in the south-east of England?