Difference between long run coefficient and non stochastic steady state coefficient ARDL model
yes, the term that you showed for the ALDR non-stochastic steady state:
$$\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$$
is long-run multiplier or sometimes also called long run equilibrium coefficient (see Verbeek 2008 Guide to Modern Econometrics 4th ed.  pp 340). As  far as I can understand there is not much difference between the two concepts in the context of ARDL model and they are used interchangeably.
The term non-stochastic steady state has wider application especially in macroeconomics (e.g. in growth theory), but often steady state results are just called long-run equilibrium because in most models steady state is attained only in long-run equilibrium.