Why are there some instances when using the second order integrated rated law that have a coefficient with the rate constant, and some don't? How do I identify when to use a coefficient on k in such cases?

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Image 1 and Image 3 are each questions regarding second order reactions, that have a single reactant type with stoichiometric coefficient of 2. I solved Image 1's question based on the guidelines from Image 2. Image 3's problem was solved using the same integrated rate law, but without the coefficient 2 on the rate constant k and I'm not sure why the rate laws are modified in each case when the problems both involve one reactant type, and the same stoichiometric coefficient on that reactant. Why are these cases solved differently, and how do I know when to use each form of the law?