I'd be more apt to say weak battery or link to battery. Is that PC625 (Odyssey, right) rated with as much cranking amps as the stock? Are you also using a charger that is optimal for that type of battery? If not, then it's longevity and reliability won't but any better than a regular battery.
Far as why it helps when you're in gear and you push is that even though the clutch is pulled, there is still drag between the transmission and crank since the plates are not completely separated, so when you help push it, you're easing the strain on the starter motor.
If jumping with a different battery don't make a big difference then I'd say a new starter motor is on order.