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The "global" refers to the fact that it works globally, not that it exists for the good of humanity.

Navstar (owned by USDoD, what is commonly called "GPS"), GLONASS and soon Galileo all have global coverage.

Yup, GLONASS = GLObal NAvigational Satellite System.

Although one should note that name is the intention; it has not always had global coverage (it does now), and may not always have global coverage in the future (say, if they decide to increase the precision in a strategic area at the cost of loss of coverage in others).

The generic term is Global Navigation Satellite System (GNSS). GPS is one type of GNSS.

I really don't see a problem with describing any and all systems that tell you your position on the globe as "global positioning systems".

Japan has its own positioning satellites (QZSS) that are only visible from the region around Japan. It’s designed so positioning works if you’re in a city with skyscrapers.