1) Have a first-aid kit on hand should you cut yourself. = Have a first-aid kit on hand in case you cut yourself.
Take heed that (1) doesn’t mean - Have a first-aid kit on hand if you cut yourself.
2) Contact an exterminator should you see fire ants in your yard. = If you see fire ants in your yard, contact an exterminator.
Hence we have the “should” in sentence 1 meaning – in case, and the “should” in sentence 2 meaning – if.
Do you agree with the correctness of my analysis? How do we distinguish the difference in meanings? Do we do it only with the help of the context? Or are their some grammatical tools (like syntax etc.) which can help us?