Ugh, why does everybody seem to be having alternator/battery problems, these days?
First, you seem to say that you have replaced the battery (is that right?), what type of battery do you have? Does it say 'silver' or 'silver calcium' on it? If not, you've got the 'wrong' battery, but in a way, you've lucked out.
That is, the older batteries are not intended for the 'smart charge' system that is on modern fords, but you can turn your smart charge system into a non-smart system by leaving off the additional connector. the alternator then defaults back to 'dumb' and 'dumb' is exactly what an old battery expects.
For 'dumb' you need three connections; an earth, a high current charging wire and a line from the dash light to the alternator field connector.
The earth connection may or may not be a wire; quite often the alternator picks up eath via the engine block and that picks up earth via a big wire (also has to take the starter current, so it will be a really thick wire) back to the car bodywork, probably in the close vicinity of the battery, so there may not be a wire connecting the earth to the alternator. If there isn't a wire, it is quite possible that due to a bad connection from the alternator to the engine or the engine to the car body, there is a voltage drop and that voltage drop would reduce the charging of the battery, which would not be a good thing.
Out of the two other wires in the 'basic' system one ought to be thicker than the other; often there are two spade connectors and the bigger spade connector is for the heavier wire. This is the line that takes the higher current back to the battery; given that, over time, this current has to balance the current taken, this can be quite a high current. In this simple system, this wire is also used for the alternator to sense the battery voltage, and if the battery voltage really climbs up, as sensed on this lead, the charge current reduces.
You have to have the charge lead and the earth connection, or you wouldn't be getting any charging, but you could have a high resistance in one of these leads, and that might reduce the charging quite a lot, but you'd still get some charging and you might be able to survive that, although it would give the battery an unnecessarily hard time.
The system with the lamp actually does two things with one connection: when the alternator isn't working (engine at 0 rpm, for example), nothing is coming out of the alternator, and the lamp comes on, warning you if the alternator is not working. The second action is slightly more subtle; before the alternator starts, there is no energy for the field winding, and so the alternator doesn't start (...this isn't quite true, but it doesn't start when it should...) and because the alternator doesn't start, there is no energy for the field winding.
Now, the current through the lamp does provide the energy for the alternator field to start up, so that cures one problem, and the fact that the alternator starts up raises the voltage on this pin so that the light goes out.
Now, if you have the smart system (which you really should have, for one of the higher-spec Silver/Calcium batteries), you will have those connections, plus some extra.
The first is a simple comms line (at least on a Mk IV); this is used so that the a module in the car can turn up the voltage (slightly) so that battery charge can be replenished more rapidly after a high load, and charging can be increased in foot-off deceleration.
The next connection that you should have is an extra battery connection; this allows an accurate reading of the battery voltage, not corrupted by the IR voltage drop in the high current connection. It ought to be easy to spot this because it has the battery voltage on it (and the system ought to work, albeit less accurately, without this connection).
There may also be a secondary earth connection back to the battery, probably a green- or a black-coloured wire. It should be easy enough to check with an ohmmeter that one of the connections goes back there
This should only leave the 'adjust' wire, and that'll be the other one!