Best-Case Running Time For Binary Search Tree Insertion
Both! That is, if you are lenient with what "best-case" means. The best-case is the input for respectively run¹ of an algorithm which minimises runtime (for a given size). I guess that some of your sources refer to (asymptotically) optimal BST implementations, either them or you mixing up the notions. 

These are the facts:


For any (reasonable) binary search tree implementation, the best-case insertion time is certainly $O(1)$ (for all sizes): all nodes are in the root's right subtree, the one to be inserted belong in the left.
An optimal binary search tree implemenentation has worst-case insertion time in $\Theta(\log n)$; it is height-balanced (examples include AVL- and Red-Black-trees).





That's equivalent for deterministic algorithms; for nondeterministic ones you consider runs.