rate of flux linking & rate of flux cutting

Yes, they mean the same. It's only looking at the same physical phenomnenon from different aspects, and is used in different situations.
Suppose when a conductor or a coil is stationary in a changing magnetic field, the [rate of flux linkage] concept is usually employed (though you still could use the [flux cutting] concept, but which is more inconvenient).
In situations where the conductor or coil is moving in a steady magnetic field, the [flux cutting] concept becomes more convenient to use.