U.S. is the preferred abbreviation of United States in American English. U.S.A. is the preferred abbreviation of United States of America, which is much more common overseas. Domestically, native speakers only use United States and U.S. in most contexts. U.S.A. only occurs for nationalistic emphasis, like at political rallies or sporting events.

Conventional abbreviations for countries and states in English aren’t defined by governments. Instead, dictionaries and style guides like the AP Stylebook (yes, the one that now includes guidance about a certain gulf) and Chicago Manual of Style include rules about these abbreviations and when to use them. These style guides differ on whether to include periods or not, even depending on context.

Within the U.S. federal government, the Government Publishing Office has its own GPO Style Manual, which specifies U.S.A. as a conventional abbreviation for United States of America, but notably the guide itself consistently uses U.S. throughout as an abbreviation of United States, even though it isn’t listed. Nevertheless, this style manual is only used in legal and administrative publications. [1]

The GPO dropped conventional state abbreviations decades ago in favor of ISO/ANSI/FIPS/USPS codes. On the state boundary relations, we’re putting those codes into dedicated keys or ref=*, reserving short_name=* for conventional abbreviations.

Wikipedia has a good overview of the various authorities for English style. [2]

I was considering swapping the short_name and alt_short_name keys, but I have no idea which one users are more likely to search for. Mainly I just wanted to correct the “U.S.A” which was missing a period. Unlike in some other languages, the last letter of an initialism needs a period too. But now that you mention it, we should list both U.S. and U.S.A. in short_name, since I don’t think 
alt_short_name is a recognized key.

[1] https://www.govinfo.gov/content/pkg/GPO-STYLEMANUAL-2016/pdf/GPO-STYLEMANUAL-2016.pdf#page=261
[2] https://en.wikipedia.org/wiki/Wikipedia:Identifying_and_using_style_guides