Those pages are not very clear indeed, but digging further into the documentation revealed the meanings of ‘local’ and ‘html5’ engines:
local = Local Storage – DOM Storage,
based on the HTML5 Specification for
local storage.
[link obsolete]
html5 = HTML 5 – Structured
Client-Side Database Storage, based on
the HTML5 Specification.
[link obsolete]
[note: the provided links are now obsolete; read more on W3.org about Web Storage and Web SQL Database.]
(source; as mentioned on the project's main page, docs are currently available on http://twablet.com/docs.html?p=jstore for the time being.)
About local storage in Chrome: that's not true; the table you linked to only point out the engine used by default in a given browser:
Does the --X-- denote "Supported" or
"Not Supported"?
Neither, the
"--X--" denotes the default browser [sic! – they actually meant ‘engine’ here]
which will be used in the given
browser.
A table of engines supported by several browser is on the Supported Browsers page. As expected, it says that current WebKit-based browsers (the table seems to be old) support all engines (except of course the IE proprietary one).