If the starter in question was on and working before, then it can't be the wrong one. There are two different ones, 6 volt and 12 volt. The number of teeth on them is different and you can't mix the two around without changing the flywheel to match. The 12 volt starter has 9 teeth on it. It should also have a red tag, the 6 volt ones are black.
If you disconnect the solinoid (and power)from the armature, and move the lever by hand, you should be able to see from underneath wheather or not its engaging the flywheel properly.
Generally, when you engage a starter and all you get is noise, either something is worn alot, or you have a bad connection somewhere. Assuming the solinoid is good, then I would go over all of your connections, especially the battery, and make sure they are all clean and tight. If all this is good, and the teeth will engage the flywheel by hand, then I would think that possibly the shaft bushings are out of round and/or the brushes are shot.