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The EMF $\mathcal{E}$ in a closed circuit $C$ is defined as the closed line integral of the electric field ${\vec E}$ around the loop: $$\mathcal{E}=\oint_C{\vec E}\cdot d{\vec\ell}.$$

But in Section 7.1.3 of Introduction to Electrodynamics by David J. Griffiths (3rd edition), in the context of a closed rectangular loop of wire, which is moved in a static, uniform magnetic field, the emf $\mathcal{E}$ is defined as $$\mathcal{E}=\oint_C{\vec f}_{\rm mag}\cdot d{\vec\ell}$$ where ${\vec f}_{\rm mag}$ denotes the magnetic Lorentz force on the charge carriers in the wire. Between the two is the correct?

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What is the correct definition of emf in a circuit?

Neither of the equations you wrote is the definition that Griffiths provided for EMF. His definition of EMF is in Eq. 7.9 of the reference you cited (his EM book).

You can spot his definitions because they use the $\equiv$ symbol.

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