<That means? That even a long time ago the martian core wasn't spinning fast enough to generate a strong magnetic field to begin with; and due to gradual cooling of the martian core, the weak martian magnetic-field grew weaker and weaker until it died.

There are regions on Mars with roughly 20 times the bulk magnetization of any similar crustal anomalies on Earth. Partially this is due to the differing geochemical and thermal history of the two planets.

What it says for certain is that for at least a short period of time, Mars had an extremely strong planetary magnetic field probably from an internal dynamo.

The simple angular momentum explanation is not sufficent to explain the phenomea of planetary dynamos. It's not a solid ball spinning away, it's magnetohydrodynamic and chaotic and the best we can generally do is deal in averages of measurable values.

You're right about cooling having an effect but it's much more complicated.