Hi Chawk! That's not all the story but it sometimes gets a guy on the right track.
Worldcat,
I give you four stars for what you've done already. Normally, I would tell a guy that, just because it's a new battery doesn't mean it's a good battery. But, you seem to have more than checked and cross checked that issue.
If jumping the spade tip on the relay causes the starter to crank then I'm sure Mike would tell you that would mean it COULD be the relay but not necessarily ONLY be the relay. It tells you that the HIGH CURRENT portion of your system is ok and the problem lies in the CONTROL section.
You see, you are working with two separate circuits when you start the engine.
You turn on a key or switch (or...keyswitch for lack of a better term), which is wired to a button (or a set of contacts in the switch), which is wired to the start solenoid and the solenoid is either bolted or wired to ground. That is one circuit. It is called the low current circuit by the engineers that came up with it. It is like this to keep human beings from coming into contact with high current and to save huge quantities of large wiring and giant switches. This "controls" the operation by simply creating a temporary electromagnet (the solenoid) to cause a set of high current points inside the solenoid to come together and create....
.... a path for current to flow from the battery, through the large cable, through the starter motor and then...and this is really important... and what trips a lot of us up quite frequently...BACK THROUGH GROUND to the battery. This is the other, COMPLETELY SEPARATE circuit. OR...CIRCLE! You MUST think of all electrical circuits as a CIRCLE that cannot be broken or there is no work. To ignore the ground side of the system is to set yourself up for failure to find the problem almost 75% of the time.
For some reason, we all, myself included, have a sort of mental block that we must overcome that leads us to think only about the POSITIVE portion of the CIRCLE. But, we cannot ignore the ground or it just will not work.
Anyway, if you jump the spade and the engine cranks, you have a control circuit failure and it should be fairly simple to go back and check:
The power to ignition.
The power from ignition to separate start button or switch (if it has one)
The power from momentary device..button-switch-key switch-whatever to the solenoid
The solenoid itself..usually done with an ohmmeter.
AND last but certainly not least, the ground than needs to be provided for the solenoid. If it is a separate wire, then that portion of the circuit needs to be scrutinized for good, clean connections and wire integrity. If it is through the contact of the mounting surface to the starter, then you are looking for paint or grease or anything that might form an insulator between the two or contamination in the mounting bolt threads.
You may use the "voltage drop" technique at each of these components and connection interfaces to isolate a high drop item or area. Note that switches and buttons are allowed up to .050vdc drop but I prefer that they drop no more than .025vdc or I replace them.
The electrical law (Kirchhoffs law) governing voltage in a circuit says that all of the voltage applied to a circuit must be dropped in a circuit speaks mostly to the LOAD in the circuit. If all things were perfect then ALL 12.6v in this circuit would drop across the solenoid as it is the item, or load, we are implementing. But, as I pointed out earlier, some voltage drop is expected throughout the circuit to allow for switches, buttons, connections and wiring. But, a maximum of 1vdc is all that is allowed for any circuits ancillary wiring.
So, if you do any voltage drop tests, write down the results of each test. Remember to activate the circuit before testing. Add all of the results of your test and if you find more than 1vdc total drop in that circuit's wiring, know that it may not work reliably.
Also, PLEASE, PLEASE, PLEASE, know that one of the real beauties of the voltage drop test is that it works for the ENTIRE circuit. That means that you can use it to find faults in the ground side connections and cabling just as easily as you can on the positive side. DON'T ignore this! As stated before, ground is just as important as power.
I hope this helps you find your gremlins.
Disclaimer: this was all written while my 5 year old grandson had some business he wanted to discuss so please excuse any spelling or typos.