There is a difference between algebraically finitely generated and topological finitely generated Hilbert $C^*$-modules. 
Consider  $C_0((0,1])$ - the $C^*$-algebra of all continuous functions on the left-open unit interval with the usual metric topology. It is a Hilbert $C([0,1])$-module as a maximal ideal of that $C^*$-algebra. 
By the Theorem of Weierstrass the finite polynomials in one variable are norm-dense in $C([0,1])$, in a modular sense the generating set $\{x(t)=t\}$ suffices to generate $C_0((0,1])$ as a $C([0,1])$-module topologically. 
However, $C_0((0,1])$ is not projective and not algebraically finitely generated.
Kasparov subsummized this kind of Hilbert $C^*$-modules among countably generated ones.