Are there some minimum criteria for a field to be registered with the FAA as an airstrip?
Good question.

14 CFR Part 139 requires FAA to issue airport operating certificates to airports that:


Serve scheduled and unscheduled air carrier aircraft with more than 30 seats;
Serve scheduled air carrier operations in aircraft with more than 9 seats but less than 31 seats.


Also:


  VFR charts depict airport's hard and soft surface runways. Enroute charts depict airports having hard surface runways of at least 3000-foot in length. If your airport meets those criteria, you may contact the local Federal Aviation Administration, Airports District Office or Aeronautical Information Management, AJV-21, at 1-(866) 295-8236 to have your airport charted.


That means for a private airstrip, it's your call as the owner to have it placed on a chart. Pretty cool.

More: https://www.faa.gov/licenses_certificates/airport_certification/