Life In 19x19

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.

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.