The Tc suppression by Cu-site substitution has been studied in a wide range of carrier concentrations p, from the underdoped regime to the overdoped regime, using sintered polycrystals of Bi2Sr2Ca1-xYx(Cu1-zMz)2O8+ with x=0 and x=0.3 and under variation of. The carrier-dependent strength of the Tc suppression by substituting M=Fe, Co, Ni, Zn for Cu is measured. In the overdoped regime, we find Tc scaling as Tc(p,z)/Tc(p,0)=g(z) with a p-independent function g(z), whereas in the underdoped regime Tc(p,z)/Tc(p,0) is strongly p dependent. Replotting data of various publications on the La2-xSrxCuO4 system and comparison with the YBa2Cu3O7- system demonstrate the universality of this distinction between the two regimes in p-type high-Tc cuprates. The scaling behavior is discussed with respect to Born-limit and unitarity-limit scattering.