The symptom you're describing could be any of the things you have mentioned, including battery. In fact, I think the most likely problem is the battery. A weak battery will power lights, radio, etc and still not be able to turn the engine over. You were not wrong to check the quality of the battery connection, that is also a likely culprit. Thestarter motor is another possibility, as well as the starter solenoid (on many cars the solenoid is integrated in the starter motor).
Check battery - To solve your mystery I would first try a jump start with jumper cables. If you do not know how to hook up jumper cables you might want to find a buddy who does to assist you. Improperly connected jumper cables can release fire, brimstone, explosions, and destroy expensive electronic components in both the jumped vehicle as well as the jumping vehicle. Use good quality cables with wire no smaller than 6 Ga (bigger numbers = smaller wire), and make sure connections are clean. There should be some, but not a lot, of arcing when the last cable is connected. You can leave the jumping vehicle running to improve the likelihood of a good jump, but you run a small risk of damaging the alternator on the jumping vehicle. If this starts your vehicle the battery is probably dead, but don't rush out and buy a new battery. Car batteries are rechargeable and being dead does not automatically necessitate a new battery. A dead car battery can be caused by leaving things turned on when the vehicle is not running or a failed alternator/charging system. Many major chain parts stores such as Advance Auto and Autozone will test batteries and charging systems for free. Drive to one of these establishments and have your battery and charging system tested before you do anything else.
Check starter - If you cannot get your vehicle to start by jump-starting then the problem may be the starter motor. If you know where your starter motor is, climb under the car and beat on it a little. Then try starting the vehicle. You may also do this with an assistant. Make absolutely sure you take excessive safety measures to prevent the vehicle from rolling once it is started, especially if it has a manual transmission. While you or the assistant turns the key the other climbs underneath and taps on the starter motor with a hammer or a wrench. If this starts your vehicle you definitely need a starter motor. Beating on the starter motor will not work forever.
Check solenoid - Many vehicles, older Fords being among them, have a starter solenoid mounted near the battery. The easiest test for your starter solenoid is also wrought with hazards. Wear safety glasses, shift manual transmissions to neutral with the parking brake set and wheels chocked, double-check an automatic to make sure it's in park. Take precautions to protect yourself from an exploding battery--both shrapnel and sprayed acid! Also pay attention to parts that may move (belts and fans especially) if you value your fingers. This sounds hazardous because it is, but this is a test I've done on my own vehicles many times, so far without incident. Now you can use a large conductive object, like a wrench, to connect the two terminals of the starter solenoid together. Keep your fingers out of the contact point because there will be hot sparks from the terminals! Also, do not allow the wrench to touch anything else! If the solenoid is the cause of your problem then the engine should crank but not start. If you do not know what a starter solenoid is, get your crazy car buddy to do this test for you. Contrary to popular belief, the voltages present in a typical automotive electrical system (assuming your vehicle isn't a hybrid) will not cause electrocution.
