1. Smart key not being detectedA Lexus with push-button start will not crank if it doesn't see a valid key. Check: Does the green "KEY" indicator or any "Key Not Detected" message appear? Try holding the Lexus emblem on the key fob directly against the START button while pressing the brake and starting. If the key fob battery is dead, this bypass method usually still starts the vehicle. 2. Brake pedal switch failureThe RX350 must see the brake pedal applied before it will crank. Check: Do the brake lights come on when the pedal is pressed? Does the START button change behavior when pressing the brake? If the brake switch has failed, the car may power up but never attempt to start. 3. Starter relay, starter solenoid, or starter motor failureThis is very common after a battery issue because the dead battery masks the real failure. Try: Have someone press START while you listen under the hood. Any click from the relay box? Any single click from the starter? No click at all suggests a control issue. One click suggests a starter/solenoid problem. 4. Main fuse blown during battery replacementWhen changing heavily corroded terminals, it's possible to short something accidentally. Check under-hood fuses, especially: AM1 AM2 ST (starter) IGN Main fusible link near the battery A blown starter or ignition fuse can cause a complete no-crank condition. 5. Loose battery connection despite seeming tightYou mentioned the terminals are "as tight as I can get them and are loose on the battery." That statement concerns me. The terminals should be: Completely seated on the battery posts. Impossible to rotate by hand. If a terminal can move or rotate, the vehicle may have enough power for lights and screens but not enough current for the starter. I'd re-check: Positive terminal fitment. Negative terminal fitment. Ground cable connection where it bolts to the chassis and engine. A quick testWhen pressing the START button with the brake pedal depressed: What exactly happens? "Key not detected" message? Green light on START button? That answer will narrow it down significantly. One thing stands out from your post: "the terminals are loose on the battery." If that's literal, I'd make that the first thing to fix before chasing anything else, because a loose connection can easily cause a no-crank condition even though the dash powers up normally.