first:
there is no such animal, you need two wires (paths), to make a complete circuit,
even a cars coil, (which is a transformer) is actually a two wire (paths) unit, the high voltage wire running to the spark plugs and the case of the coil which attaches to the bock of the motor and the spark plugs are also attached to the block of the motor, thus two wires even tho you only have (one visible wire) in other words there needs to be a complete circuit, which takes two avenues of path, (wires),
there is two wires,(paths) in the primary side of the transformer, and there are two or more paths, (depending on the taps) on the secondary side of the transformer), but each voltage needs a complete circuit to travel thus (for sake of better description) a path out of the transformer coil and one back in to coil. Now like in the cars coil the case of the coil is the other path (wire),
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two: apparently the transformer you have is different than the one that is in the generator, and you may need to go and get actual parts for that model
three: it sounds like to me you may be wise to take it to a an actual repair place that understands the workings of the generator and get fixed and checked out by some one who understands what is going on in the unit,
if you get it hooked up wrong you may end up costing your self much more than the price of the transformer,