A direct air carrier must obtain an air carrier certificate. Any entity operating as other than a direct air carrier must obtain an operating certificate.
https://www.ecfr.gov/cgi-bin/text-id...se14.3.119_139
https://www.ecfr.gov/cgi-bin/text-id...3.110&rgn=div5
"Airline" doesn't really have a meaning in the regulations/FAA. The certificate holder is either an operator or an air carrier.