About terminology:
We call strings like "America/Denver" etc. (in format region/city) timezone identifiers, not names. These ids are defined in IANA-TZDB. Timezone names and abbreviations are a totally different concept. Those data usually have their origins in CLDR-files (from Unicode consortium, for example here) and correspond to what most people expect IN THEIR LOCALES. These names are highly localized. In general, there is no international standard, neither for identifiers nor for names. Latter ones are just following local customs.
Mapping:
A mapping from timezone identifiers to country codes and vice versa can be found in IANA-TZDB. You can then quickly see that there is no unique mapping as you expect. Examples:
Europe/Zurich can be valid in Switzerland, Germany(sic!) and
Liechtenstein.
Or trivially:
US has more than one timezone (America/New_York, America/Chicago,
etc.)
Your hope to get unique timezone abbreviations using the expression
Calendar.getInstance().getTimeZone().getDisplayName(false, TimeZone.SHORT);
is not realistic. The method to get tz abbreviations you describe is correct. However, as you yourself have noticed in your other SO-post the results differ from one Android device to another. The main reason is that different devices (dalvik vms) often have different data for timezone names (and the OpenJDK-Oracle-VMs, too).
A unique mapping from timezone names and abbreviations to country codes is usually not possible. Counter examples: "IST" can stand for "India Standard Time" or for Israel time zone. Or: "BST" is used in "Europe/London" (during summer time) or in "Pacific/Bougainville" (all the year).
About daylight saving:
Identifiers are NOT sensitive for daylight saving. The related data can be daylight saving or not, dependent on the associated time. Timezone names and abbreviations are often (not always) sensitive, compare for example "PST" or "PDT" in US.
Summary and advice:
Do not use timezone names or abbreviations for anything other than display purposes. You should not use these localized data for any conclusions or storage purposes.