Metropolitan area boundaries notes
By checking timetables, the stops that the vehicle will take, and whether the message is present, you can see whether you're inside the metropolitan area, or outside, or about to head into a metropolitan area.
This is mainly intended for geeks or folks who are interested in figuring out which stations are in a metropolitan area. For folks who are wondering if they need more than a Budapest-pass to travel into certain stations, the messages "Budapest-passes are not valid on this section." and "The validity limit of Budapest-passes is marked by a pink line." means you'll need and probably need something more than a Budapest-pass respectively.
The boundaries for Budapest passes is marked with a pink line in between the stops.
For bus timetables:
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"Budapest-passes are not valid on this section." means that you're currently inside the metropolitan area (https://bkk.hu/apps/menetrendek/storage/menetrendek/2171/494457.pdf). However, the same message also shows up if the next station of the line is going to enter the metropolitan area (see 2 stations before metropolitan and 1 station before metropolitan).
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"The validity limit of Budapest-passes is marked by a pink line." means that you're about to head into a metropolitan area. https://bkk.hu/apps/menetrendek/storage/menetrendek/2171/494480.pdf
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If there's a pink line but no message about metropolitan boundaries, that means you're out of the metropolitan area and you are heading out of the metropolitan area. https://bkk.hu/apps/menetrendek/storage/menetrendek/2171/494458.pdf
For H5-H9 timetables:
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H5 has a pink line, but no message
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H6 has the same form as H8/H9
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H7 has seemingly no boundaries, possibly because it does not cross outside of Budapest
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H8/H9 has a very bright pink thing saying which station is still valid for Budapest-pass, but says pretty much nothing else
For trams: There are no trams which cross metropolitan areas.
Finding vehicles that go into metropolitan areas
So far, there is Metropolitan area and transport hub maps, however that does not include all vehicles that go into metropolitan areas.
There is https://bkk.hu/en/journey-planning/maps/daytime-public-transport/ which links to https://bkk.hu/downloads/map/171/ (warning: 18MB PDF!), however it's a pretty large PDF and rendering it to an image (using GIMP, rendering it at 300 PPI, and exporting it as a compression level 3 PNG) makes the PNG over 170MiB in size.