When you say it leaks while using the shower I'd like to advise you that a small amount of leakage out the spout is normal. It is this function that causes the diverter to release after use so the next user doesn't catch a surprise. I have a tight one presently installed at my house and I can't wait for it to loosen up.
That said if the leakage out the spout is excessive or it is leaking behind the spout replacement is in order. You are correct there are several types either screw on or, slip on. If you do not see an allen screw up under the slot it is probably a screw on or, it is a slip on that got spun by a gorrilla. When I remove a tub spout I usually take a pair of channel lock pliers and break off the outer shell. This enables me to hold back on the nipple or stub out and prevent it from breaking off inside the wall. It also enables you to see the manner of attachment.