Can you go into a bit more detail about exactly what it does when you try to start it on the key?
Traditional starter motors need a small current from the battery to the key to the starter's solenoid. This is essentially a combined relay that turns power to the starter also while pushing the starter motor's sprocket into the flywheel's matching teeth and spins the engine over.
When you release the key the starter solenoid powers off, disabling the starter motor, freeing the flywheel cog, and allowing the main motor to go faster. Cranking speed might be around 100 RPM whereas engine idle speed for a car is about 1500 when cold down to 800 RPM when warmed up.
Upshot, if you get a click/clonk noise followed by silence when you turn the key, the starter is not engaging at all, and it needs to be visually inspected for loose/dirty wires and connections in place. If they all look good then pull and rebuild, swap, or replace.
If you get a clunk and the starter motor then begins turning, but its really weak and slow, then you're not getting the current needed. The starter running is the moment of highest energy drain the battery will see, so it needs to be pushing 300-700 Amps.
Look at the wiring for cracks, damaged bits, burned insulation. Check the big fat cable from the battery's positive to the starter, AND check the earth strap from the engine to the chassis and the one from the battery to the chassis.
That the car starts when jumped to a donor vehicle suggests your battery is weak and the starter motor is fine. The battery might simply be low rather than weak, which puts the blame on your alternator for not topping it up while driving.
Aside: You can possibly leave your car on a trickle charger overnight and see if it starts by itself in the morning - if it starts fine then your alternator isn't doing it's job. If it still doesn't start after overnight trickle-charging, then your battery is potentially EOL. You can check the date code on the side of the battery to see how much the seller lied about "a new battery" but do consider modern starter batteries only last for 3~4 years before needing replacement again.
Modern fancy cars also have computers and safety interlocks to save the operator from bad decisions (like starting when in-gear) It may be possible one of these is stopping you, that is not active when the car is being jump-started. Fairly unlikely but not impossible.