How should private airports or locations not regularly intended for landing be identified if you land at or take off from them?
In the USA, as far as your logbook is concerned, FAR 61.51(b)(1)(iii) only requires "Location where the aircraft departed and arrived," not the 4-letter ICAO identifier of your airports. When you abbreviate the airport with its identifier, that's all you're doing: abbreviating. You could, for instance, write "Nashville International Airport" instead of "KBNA."

Seaplane and helicopter pilots often land in places that aren't even airports. You might log the destination as "Murphy Lake" or "Walmart Parking Lot." Such is the standard convention.

For purposes of the flight plan, there are a few options if flying to a location that has no identifier. On an ICAO flight plan to a location with no identifier, you would enter ZZZZ as your airport, and in the remarks section include a DEST/ group including the lat/long of your destination. For instance, I think DEST/3607N08641W would work. On an FAA flight plan, you can list the nearest waypoint to your destination, and again, describe your actual destination as a common name or lat/long in the remarks.