Relation between logspace-uniform circuits and P-uniform circuits
The circuit value problem (CVP) is, given a circuit and an input, to evaluate the circuit. This is a well-known problem which is P-complete with respect to AC$^0$ reductions. Thus one can define P as the class of problems AC$^0$-reducible to CVP. We get the same class if we consider the class of problems polytime-reducible to CVP, or in general $L$-reducible to CVP, where $L$ is any intermediate class between AC$^0$ and P. An $L$-reduction to CVP is very similar to an $L$-uniform circuit (the only difference is that the inputs to the circuit could be any $L$-functions). Specifically, an AC$^0$-reduction to CVP can be converted to a uniform circuit (the notion of uniformity depends on the exact format of circuits, but it can probably be taken all the way down to AC$^0$).

The same phenomenon occurs whenever we have a universal circuit (in the case of P, this is the circuit solving CVP). For a recent example, see this paper on comparator circuits.