Car will not start

I assume the symptom is the starter does not crank the engine. What type of transmission do you have? Testing a stater motor off the engine is not a valid test. What do you hear or observe when the problem occurs?

That is a totally different symptom than a no-crank. You still did not say what kind of transmission you have, but it is irrelevant now. A real good suspect with an automatic is a bad neutral safety switch/transmission range sensor. The symptom would be "no crank".

Slow cranking means all of the light-current circuit, to include the ignition switch and neutral safety switch is working. We know that because the starter relay is tuning on. Next, we know the medium-current circuit is working because the starter solenoid is engaging. That includes the starter relay, starter solenoid, and its wiring.

That leaves the high-current circuit which includes both fat battery cables, their connections, the contacts in the starter solenoid, the starter motor, and the battery. We can skip the battery since you have a new one.

There has to be a high-resistance connection in the high-current circuit. That resistance is way too small to measure with an ohm meter, but we can measure the results of that resistance with voltage drop measurements. Start with the voltmeter on the 20-volt scale, then put the two probes right on the battery's posts. You must find 12.6 volts there. Measure that while a helper tries to crank the engine. Another, less-accurate test is to see how much the head lights dim during cranking. The voltage must not drop below 9.6 volts. If it does, the engine is tight or has severe drag. If the voltage stays close to 12.0 volts or higher, the starter is not drawing enough current, and that is due to a high-resistance connection.

If the voltage stays high during cranking, move the probes from the battery posts to the cable clamps, and try again. You should get the exact same voltage reading as when the probes were on the posts. If the voltage is noticeably lower, one of those connections is loose or dirty. You can determine which one by moving just one meter probe at a time.

If the second test was okay, move one meter probe from the negative cable clamp to a paint-free point on the engine, then try again. If the voltage is a lot lower during cranking, that cable has a bad connection. The last step, if you can get to the starter easily, is to move the positive probe from that cable clamp to the larger stud on the starter solenoid. That must be on the copper stud or nut, not the terminal attached there. If the voltage reading is lower now, that cable has a bad connection.

For both battery cables, be aware corrosion can occur in the first inch under the insulation where it cannot be seen. When enough strands of the cable are corroded away, you will not be able to get enough current through to the starter. That shows up as the lower voltage readings during cranking.

12.6 volts means the battery is fully-charged. 12.4 volts means it is considerably discharged.

The industry accepted standard for battery voltage during cranking is a minimum of 9.6 volts. To have less than that, the battery is weak or the engine has too much drag. Since the battery was already tested, the next test is to see how much current the starter is drawing. A load tester is needed for that.

This can get confusing because the starter motor is actually two motors in one housing. It draws real high current to get started spinning, and that draws battery voltage down a lot. Once the motor is up to speed, the amount of current it draws drops a lot, and that lets battery voltage go back up. Where this gets confusing is if one of the four brushes inside the starter motor is worn excessively, only one half of the motor will work. That makes it spin too slowly and it never gets up to a high enough speed where current will drop down.

Normally the clue is starter current appears to be normal, but cranking speed is too slow. To say that a different way, starter current remains too high because it has not gotten up to speed, but it is half of normal because half of it is not working. The best way to identify this is to replace the starter, but the clue is battery voltage stays well above 9.6 volts. I know you had the battery tested, but it still looks like it is weak.

Try rotating the engine by hand to see if it's tight. Next would be to use a professional load tester to see how much current the starter is drawing. Maximum should be around 150 amps, then it should drop down once it's spinning. You can also get an idea by watching the amp gauge on a larger wheel-type battery charger. Check if the battery voltage during cranking stays higher with the charger connected. 12.4 volts means the battery is not fully-charged, but it's not dead either.