Storing timezone in a database
The timezone strings like America/Halifax and Asia/Kolkata refer to entries in the so-called zoneinfo data base. Read this. https://www.iana.org/time-zones It's presently maintained by the Internet Assigned Numbers Authority. It changes a lot, because its maintainer scrambles to keep up with changes to daylight saving rules and other temporo-political stuff.
On UNIX-like systems the entries are stored in a filesystem directory hierarchy, and not by some code number. That is a successful implementation of my suggestion. You can take a look to see how it works.
You asked:
Do I store the whole
America/Los_Angelesstring
Yes.
is there some sort of integer ID for it?
No, not if you want your application and data base to be future-proof. There is a time_zone_id in the part of MySQL that holds the zoneinfo data, but nobody has made any promises about keeping those numbers unchanged.
The longest string in the current (2021e, still current as of March 2022) data base is posix/America/Argentina/ComodRivadavia. It's 38 characters long. The mysql.time_zone_name table has a Name column defined with 64 characters. It makes sense to use VARCHAR(64) for storing this information; that matches the way the names are stored in the system. By the way, the names are all in 7-bit ASCII, so you can use the ascii_general_ci collation to define the column where you stash them. This should be a good definition for a user preference time zone column.
time_zone VARCHAR(64)
NOT NULL
DEFAULT 'UTC'
COLLATE ascii_general_ci
WordPress offers the user a picklist (drop down list) of available time zone names, and stores the user's choice as text. It's a great example to imitate.
http://www.iana.org/time-zones
Check here MySql DateTime
MySQL converts TIMESTAMP values from the current time zone to UTC for storage, and back from UTC to the current time zone for retrieval. (This does not occur for other types such as DATETIME.) By default, the current time zone for each connection is the server's time. The time zone can be set on a per-connection basis. As long as the time zone setting remains constant, you get back the same value you store.
You should add a new column in your table to keep your timezone and when extract information convert your time zone.