Clear definition of a 'non-oxidizing acid'?
A non-oxidising acid is an acid in which there is no stronger oxidant than $\ce{H+}$ at $\mathrm{pH}\ 0$.

Or, in a slightly different wording, a non-oxidising acid would be one that cannot dissolve any noble metal at $1~\mathrm{M}$ concentration, but should (in the absence of effects like overpotential) dissolve any non-noble metal at that concentration.

Nitric acid has the strong oxidation agent $\ce{NO3-}$ present, which will reduce to $\ce{NO}$ or $\ce{NO2}$ depending on the concentration. It is able to oxidise noble metals such as silver.

Hydrochloric acid, on the other hand, has no other oxidising agent. The chloride ion is already fully reduced. Therefore, it is not able to oxidise noble metals such as copper.

Some acids are more strongly oxidising if they are concentrated. Sulphuric acid is one. Only in higher concentrations will reduction of sulphate to $\ce{SO2}$ take place and make the acid oxidising. In lower concentrations, only $\ce{H+}$ takes part in oxidations.