It sounds to me like you have a poor connection from a battery terminal (connector) to the wire that is crimped to that connector. After you crank, check those battery terminal connectors for any heat. If one is warm then you have a high resistance connection. It won't matter whether you are jumping the battery or just using your old battery. If a wire terminal has high resistance, the starter won't get enough voltage to spin it sufficiently to start the engine.
To repair, you can either buy a new cable (preferred), or cut the wire off the old cable, strip the insulation back, and install it on a new battery terminal connector.
Note that the problem could also be a short within the starter. See if the battery terminals get hot first and if they don't, remove the starter and get it tested at your local auto parts store.
To repair, you can either buy a new cable (preferred), or cut the wire off the old cable, strip the insulation back, and install it on a new battery terminal connector.
Note that the problem could also be a short within the starter. See if the battery terminals get hot first and if they don't, remove the starter and get it tested at your local auto parts store.