Considering Breadth-First Search and Depth-First Search perform a traversal operation and not a search operation
I would not say that as a blanket statement. While it does make sense in certain context it can be very misleading.
As one who programs in Prolog daily, traversal and search are very distinct.
When I see the word traversal used explaining code I expect the code to visit every node of a structure (think tree, graph, list, etc.). Traversals are typically hidden inside map functions and thus visit every node.
When I see the word search I also think of guards that modify the search and search indexes that make the search more efficient. Also a search may visit all of the nodes, some of the nodes or only one node, a search does not always need to visit every node and often does not.
However there are other types of methods/functions/predicates that use callbacks and will traverse a structure but use the call back to access guards and search indexes making the traversal more efficient.
You noted binary tree but it makes more sense if one uses a binary search tree instead meaning the values are ordered such that for a node all values greater are in the left branch and all lesser values are in the right branch.
A traverse of a binary search tree will visit every node of the tree.
A depth-first search of binary search tree will use the value at the node to decide to search the left or right branch. This obviously does not traverse the entire tree.
A breadth-first search of a binary search tree seems senseless. (confused emoji goes here) It is ignoring the fact that a binary search tree is constructed to quickly find a value by comparing the current value and picking the left or right branch.