Relaxing Semi-Honest Security in Multiparty Computation
I'm not sure that the issue here is uniform versus non-uniform, but I'll address this first. It is possible to provide a uniform definition. In order to see what it would look like, I recommend reading Oded Goldreich's uniform treatment of zero knowledge http://www.wisdom.weizmann.ac.il/~oded/PS/uniform.ps. 

In any case, your question is more about quantifying over all possible inputs, versus over a specific distribution. (Note that in Oded's uniform treatment, the quantification is over all poly-time samplable distributions. Here, you wish to guarantee security for a specific distribution only.) It is possible to define such a notion and it does make sense. The only question is if it buys you something. I have used such a notion once, in order to construct a secure password authentication protocol; see the definition in http://u.cs.biu.ac.il/~lindell/PAPERS/session-key.ps (Section 2.3.2). In this definition, the password is chosen randomly from a dictionary as part of the ideal and real models. You could do the same thing and choose a random key.