References:
forum/viewtopic.php?p=225367#p225367
forum/viewtopic.php?p=230291#p230291

"Stones are 'alive' if they cannot be captured." is very and fundamentally wrong. Formal definitions are here:

http://home.snafu.de/jasiek/j2003.html
http://home.snafu.de/jasiek/wagcmod.html
http://home.snafu.de/jasiek/sj.html
http://ljrg.eu5.net/
Mathematicap Go Endgames, chapter B.2
Olmsted's rules in The Structure of Go and The Rationalisation of Go

In particular note that normal go has three types of capturability-life: uncapturable, capturable-1, capturable-2. See also

http://home.snafu.de/jasiek/j2003com.html
http://home.snafu.de/jasiek/j1989c.html
http://home.snafu.de/jasiek/wagcflaw.html

Do any of the links directly address the definition that stones are alive iff they cannot be captured? I see that the 2003 Japanese rules state a different definition, but does not seem to state why the difference is present or in what situations the one definition gives incorrect results.

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Occupy Babel!

See the examples referred to by the other links. The most basic shape class: eyespace with capturable-1 (live) stones inside. To be learnt before snapback.

For answers to your other questions, do read the linked commentaries.