you start always with the position vector to the masses .
Case I
$$\vec P_m=\begin{bmatrix} X \\ Y \\ \end{bmatrix}=\left[ \begin {array}{c} \left( l+x \right) \sin \left( \phi \right) \\ - \left( l+x \right) \cos \left( \phi \right) \end {array} \right] $$ the generalized coordinates are $~x(t)~,\phi(t)~$
from here the velocity $$v^2=\dot X^2+\dot Y^2 $$ and the kinetic energy $$T=\frac m2\,v^2+\frac k2\,(l+x)^2$$
Case II
Position to $~m_1$
$$\vec P_1=\begin{bmatrix} X_1 \\ Y_1 \\ \end{bmatrix}=\left[ \begin {array}{c} r\sin \left( \phi_{{1}} \right) \\ r\cos \left( \phi_{{1}} \right) \end {array} \right] $$
Position to $~m_2$ $$\vec P_2=\begin{bmatrix} X_2 \\ Y_2 \\ \end{bmatrix}=2\,\vec P_1+\left[ \begin {array}{c} r\sin \left( \phi_{{2}} \right) \\ r\cos \left( \phi_{{2}} \right) \end {array} \right] $$
the generalized coordinates are $~\phi_1(t)~,\phi_2(t)~$
thus the velocities are
$$v_1^2=\dot X_1^2+\dot Y_1^2 \\ v_2^2=\dot X_2^2+\dot Y_2^2$$
and the kinetic energy
$$T=\frac m2\,(v_1^2+v_2^2)$$
What is the general principle that determines whether an independent variable should be treated as a parameter or explicitly as a variable in such problems?
the only independent variable are the generalized coordinates , thus are implicit time dependent.