A weak solution consists of a filtered probability space carrying a Brownian motion and an adapted continuous process satisfying
(1)
The probability space and Brownian motion are part of the unknown solution.
A strong solution is adapted to the augmented filtration generated by a prescribed Brownian motion and initial condition; equivalently, it is constructed measurably from that given noise.
Uniqueness in law means that any two weak solutions with the same initial distribution have the same probability distribution on path space.
Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and with the same initial value, are indistinguishable.

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