This one definition is actually quite simple, a cutter has two forestays.
A sloop has one. I'll be surprised if anyone disagrees with this .... but sailors can be crochety
Boatpoker, I think that is the modern, popular definition, but I know a few old-timers who insist that the number of headstays or foresails is only incidental; in their minds, a sloop can have two headstays/foresails as well. It is the placement of the mast that separates the cutter from the sloop. I can't remember the formula, but a cutter has the mast set further aft than a sloop does. I believe that under this older definition, boats such as the Island Packets which most of us would consider cutter rigs (because they can fly two foresails) are actually sloops.
Sabiny, you asked would a sailor use just the jib if moving from one wharf to another in the same port; the inference is that it might be easier to do so if singlehanding. I think the answer is "maybe"; it would depend on the winds and current. Sailing with just the headsail is problematic; it makes steering difficult because the rig is unbalanced. The jib is creating forward drive, but is also pushing the boat away from the wind, requiring a lot of effort to steer the boat. Imagine trying to swim forward if someone were pushing sideways on the end your outstretched arm. Steering on one of these boats requires a great deal of physical effort; the tiller or wheel is always trying to push you one way or the other. And trying to sail with just the jib is particularly difficult. You have a big sail with a great deal of leverage (remember, its at the very end of the bowsprit on the maximum lever arm) fighting you. The drive from the main sail is what balances out the force from the jib. Boats with properly designed and sailed rigs are balanced; the amount of effort that it takes to steer is minimal. Sailing with just one sail (any one sail) unbalances the boat.
All that being said, sometimes it makes sense to have just one sail up. It depends on the boat and the conditions. If you want to avoid having your sailor raise the main and use just one sail, I think he would use the fore staysail, not the jib. The jib is the triangular sail furthest out from the main. The fore staysail (sometimes called the inner jib) flys closer to the mast than the jib, and is therefore more suitable for what you envision. Certainly easier to hoist than the main sail on a gaff-rigged boat such as the one you have been describing. But if you are going for authenticity, make sure that the wind is blowing your boat off the first wharf and onto the second. Otherwise, the single-handed scenario is implausable. If you want your sailor to appear expert, I suggest you check out some of the sailing guides that give tips on how to leave docks under sail. You can do some nifty stuff with spring lines, fenders and a clever use of sail and the rudder.