Are there different terms for biosynthesis of a compound naturally in an organism versus the biosynthesis using chemicals in the laboratory or is it just "biosynthesis"?
More often than not, the synthesis in organisms involves the action of enzymes. In the lab, you often avoid these for various reasons. The enzymes may not be available, or you're interested in a more general procedure, that allows to synthesize a number of derivatives of the target structure or just a part of it.
When the synthesis aims to obtain a particular molecule, just as found in nature by any means, you'll often find a term like natural product synthesis.
When a procedure aims to yield a product by a reaction similar to processes in living organism, the term biomimetic is frequently used.