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In underdoped cuprates, the value of the upper critical magnetic field (H_c2) has been under debate. However, recent studies on the La-214 family have firmly established vortex phase diagrams that include an intermediate, viscous vortex liquid regime, regardless of the presence of charge order. We perform similar studies in underdoped Bi₂Sr_2-xLa_xCuO_6+δ (La-Bi2201), which exhibits short-range charge order, by measuring linear transport, voltage-current characteristics and Hall effect in H up to 45 T and temperatures (T) down to 0.017 K. Our data reveal an insulatinglike high-field normal state and a qualitatively the same vortex phase diagram as in underdoped La-214 cuprates, thus supporting the universality of the vortex phase diagram with an intermediate phase, i.e. the importance of quantum phase fluctuations as T→0, in underdoped cuprates. The properties of the apparent two-stage superconductor-insulator transition are discussed and compared to that in highly underdoped La_2-xSr_xCuO₄.