A "weak" pull resistor is usually a high value resistor that only allows a small amount of current through, and can quickly be overwritten, but takes longer to reassert itself.
A "strong" pull resistor is usually a low value resistor, allows more current through, takes longer to be overwritten, but can quickly reassert a line.
They are completely relative to your needs, not just other pull resistors like internal ones.
In your button scenario, the time it takes to switch from one state to the other isn't important, so weak vs strong doesn't apply there. But weak vs strong does apply in the practical matter of Current Consumption. A strong pull resistor would, when the button is pressed, cause a large drain of current from vcc through the resistor to ground. A weak pull resistor would cause a small drain of current. Theoretically any resistor would work, but for practical purposes, a weak resistor is used because unnecessary high current drains can cause issues and can easily be avoided by sizing the resistor correctly.