Only sort of. In general each town had their own time keeper, and clock. Some would adjust their clocks to sun noon more often than others, and some were stricter about doing it. So you could never be sure that the next town west had noon later than this town (though in practice mechanical clocks were so bad that you couldn't measure it)
Also in some area time was set by sunset not local noon. (Jewish cares about when the sunsets - I suspect others as well)
It's only for days of the week. The time is still the local time.
Since it's very important in Judaism to not do any work on Shabbat (the Jewish day of rest), there are two distinct Shabbat's in Israel. One is "religious" Shabbat (translated to English as "Shabbat") which starts at sunset on Friday and ends after sunset on Saturday, lasting about 25 hours. The other is "civic" Shabbat (translated to English as "Saturday"), which starts (as everywhere) after 23:59 Friday and ends at 00:00 when Sunday comes.
So when you invite a friend over on "Shabbat evening" there's an ambiguity: it's unclear whether that relates to Friday evening (the evening of "Shabbat") or Saturday evening (the evening of, ahem, "Saturday"). To resolve that ambiguity, modern Hebrew has a distinct term for "the evening after Shabbat that comes on Saturday evening".
