What makes a pull-up/down resistor strong or weak?
Strong means low resistance. Weak means high resistance. Of course low and high are relative terms, and so are strong and weak. The reference for this relationship must be inferred from context.

A strong or low resistance pull-up/down resistor is good because the time constant formed the load capacitance (often, the input gate capacitance, and the PCB trace capacitance) is small, so rise/fall times will be short.

A strong pull-up/down resistor is good because noise currents from unintended coupling and EMI will result in smaller noise voltages. (Think about Ohm's law)

A weak or high resistance pull-up/down resistor is good because it will not require much current from the driving circuitry to work against the resistor. Batteries will thus last longer, parts can be smaller and don't get as hot.

Of course, you usually want all of these things, but a resistor can't be both. A discussion about strong vs. weak is usually clarifying which of these concerns (or perhaps others) are more important for a particular application.