MatchLocalNames is an option for Trace and related functions which specifies whether symbols such as x should match symbols with local names of the form x$nnn.
The default setting is MatchLocalNames -> True.
With the default setting, Trace[expr, x = rhs] will show assignments to local variables whose names are of the form x$nnn.
Trace[expr, x = rhs, MatchLocalNames->False] shows assignments only for the global symbol x.
See The Mathematica Book: Section 2.5.10.