The ESV most likely follows the version of the LXX in which the word bow is missing from the text. In this version there are no problems with the text, it is smooth and well ordered.
The ISV and Rashi's interpretation on the other hand I think are hardly defensible: Why would the text tell us now that David taught the Israelites archery? It interrupts the flow of the lament and doesn't follow any logical or natural order and seems to me completely out of place. If Rashi were correct in his translation, v18 (archery) would've preceded v17 which introduces the lament of David; for why would the text first introduce the lament in v17 only to distract the reader with a marginal, tangential note about David teaching the people archery. This is unjustifiable. Take a look at the text yourself.!
17Then David took up this lament for Saul and his son Jonathan, 18and he ordered that the sons of Judah be taught archery. It is written in the Book of Jashar 19“Your glory, O Israel, lies slain on your heights. How the mighty have fallen!
As you can see, v18 (which I have italicized) is completely out of order and makes these verses awkward and unreadable.
If we must accept the word bow as part of the original text then the interpretation of the ASV and NIV is most likely. It is the "song/lament of the bow" that David taught to them. Indeed it is understandable why the author inserted this verse as part of the introduction to this lament.
The E. V. cannot be right in inserting “the use of,” for the bow was a weapon already in common use. If the text is sound, “the Bow” must be a title given to David’s elegy from the mention of Jonathan’s bow in 2 Samuel 1:22. (Cambridge Bible Commentary)