I think of it as the corpus of cultures drawing heritage from the Ancient Greeks and Romans. If the Homeric epics have a home in your culture, it's Western.
That said, the term is muddied enough to be somewhat useless for general discussion. Particularly given how it's been co-opted by racist elements in American politics.
Those of us who consider ourselves Western are right to consider the Greeks and Romans among our cultural forebears, but we're not the only ones who can or should do so.
Most of the reason Europe was able to even have access to Greek works was due to Arabic translations, in many cases the originals do not exist anymore.
While in Europe the barbarians were ravaging the remains of the Roman Empire and entering into feudalism and the 'dark ages', Hindu and Greek works were being studied and improved in the Arab world (e.g. Algebra, Algorythm words coming from Arabic works).
These later would be brought to Europe via trade with italian city states and copied by the church.