That definition isn't only used by libertarians, though it's possible that libertarians are more likely to use the "violence" formulation rather than the "physical force" formulation.
International recognition is possibly a separate issue.
According to some theorists, there are states that aren't recognized as states, but may still be states. They may actually rule territory (perhaps quite a stable way) but other states still may not recognize them -- most often because other states support a different claim to rule the territory that the unrecognized state rules. A familiar example in Europe is Northern Cyprus, where there is a state-like entity that has ruled a particular territory with no practical physical challenge for a long time, but almost all other states in the world support a different entity's claims to that territory and wish that the existing ruling entity would dissolve or be absorbed into the other entity.
https://en.wikipedia.org/wiki/Monopoly_on_violence
That definition isn't only used by libertarians, though it's possible that libertarians are more likely to use the "violence" formulation rather than the "physical force" formulation.
International recognition is possibly a separate issue.
https://en.wikipedia.org/wiki/Sovereign_state
According to some theorists, there are states that aren't recognized as states, but may still be states. They may actually rule territory (perhaps quite a stable way) but other states still may not recognize them -- most often because other states support a different claim to rule the territory that the unrecognized state rules. A familiar example in Europe is Northern Cyprus, where there is a state-like entity that has ruled a particular territory with no practical physical challenge for a long time, but almost all other states in the world support a different entity's claims to that territory and wish that the existing ruling entity would dissolve or be absorbed into the other entity.
https://en.wikipedia.org/wiki/Northern_Cyprus
And there are a lot of other situations like that, often with positions on the claims determined by other political considerations.