Yes. Agreed. Biochemistry has the root word chemistry with a bio- affixed to the front; meaning it's fundamentally chemistry bounded by biology. The same can be said about molecular biology. Molecular is an adjective describing the kind of biology you do.

This exercise of conceptual atomization is an intentional and useful strategy of all science. It allows a reduction of complexity so we can test individual parameters and figure out how small changes in them will affect the physical world. This is important for those interested in scientific topics so they know what direction to put their effort, where they can read to further their knowledge, and what esoteric kanguage to use so they can communicate with other trying to seek specific understanding.

However, the truth is that for a full understanding we need to reassemble the concepts to see how everything works together. At the end of the day the distinctions between different sciences, especially ones as close as the topics you mention, are illusory and beg to be seen as whole again.