That is the meaning of "activity" as a technical term, but it's obviously not what is meant by calling lithium "more active than the other alkali metals". Presumably this refers (in rather loose language) to the fact that Li has the most negative reduction potential, meaning that the thermodynamic driving force for oxidation of Li to Li+ is greater than for the other alkali metals, though not by much (I can't think of anything that could be reduced by Li but not by another alkali metal). A battery with a lithium metal anode would have a slightly larger voltage than one with a sodium anode, for example.
Reactivity is about how rapidly something reacts with something else, and this generally increases down group 1, so francium would react more violently with water, say, than lithium would.
We have hit upon the vital distinction between thermodynamics and kinetics, which runs through chemistry.
Thermodynamics is about the relative stability of chemicals/states.
Kinetics is about the rates of processes.
They are not necessarily correlated - a thermodynamically favourable reaction may be kinetically very slow.