Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by $$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$ $$[c,L_n]=0.$$

Similarly, following the standard conventions in the literature, the commutation relations of the Neveu-Schwarz super Lie algebra are given by $$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{8}(m^3-m)c,$$ $$[J_\alpha,J_\beta]_ +=2L_{\alpha+\beta}+\delta_{\alpha,-\beta}\frac12(\alpha^2-\frac14)c,$$ $$[L_m,J_\alpha]=(\frac12m-\alpha)J_{m+\alpha},$$ $$[c,L_n]=0,\qquad [c,J_\alpha]=0.$$

The Virasoro algebra is a subalgebra of the Neveu-Schwarz algebra by $L_n \mapsto L_n$ and... $c \mapsto \frac32c$. Clearly, one could change the normalizations in the second set of formulas to avoid the annoyance of having to send $c$ to $\frac32c$.

So whyyy do people do it that way?

Why do physicists take their standard formulas not consistent with each other?

There must be a reason.