The term is unintuitive to me in that usage, because to me a protocol implies an exchange or a sequence of steps between two or more parties, such as in network or cryptographic protocol or a diplomatic protocol. Interfaces, however, only specify one endpoint of an interchange, in an inherently asymmetric way, and they primarily specify operation signatures, where there are often few constraints on the possible sequence of operations.
An interface can specify or be part of a protocol as a special case, but to me the term doesn’t match the general case.
An interface representing a state machine usually does not directly represent the concrete state machine, but presents an interface to interact with the state machine. E.g. for a parser state machine, submit the next token that will advance the state machine; or an operation to read out the current state, or an operation to obtain a list of the currently possible transitions.
Of course it's not impossible for the preconditions for each operation of the interface to map exactly to a protocol so that the operations permitted in each state correspond 1:1 to the state transitions of a protocol. But that is not the general case, and not how people usually think about interfaces.
I don't know if you are familiar with the term interface description language (IDL), but generally IDLs are not suitable for specifying a protocol, in the sense of specifying (among other things) the protocol's state machine. It would be misleading to call them protocol description languages.
There are other ways in which "interface" and "protocol" differ. For example, we call HTTP a protocol, and things like SOAP and REST and CORBA. But we call a concrete REST API an interface, not a protocol, and we call what is defined by a WSDL an interface, not a protocol. A protocol can be used to access an interface (I use SOAP to access a monitoring API). An interface may support different protocols (I can operate different protocols through the Unix socket interface). We talk about Linux kernel interfaces, not about Linux kernel protocols (and if we would, we would mean something different by that). The two words are not interchangeable.
An interface describes the boundary between two things. A protocol describes a behavior between multiple entities. By the implementation of an interface we mean the implementation of one specific side, the one that exposes the interface, not the one that uses the interface. An interface can exist even if no one uses it (only one side exists). The implementation of a protocol, on the other hand, is split between both sides. You can't have only a one-sided protocol. If no one uses the protocol, neither side exists, so to speak.
An interface can specify or be part of a protocol as a special case, but to me the term doesn’t match the general case.