Connection between probabilistic algorithm and distributional algorithm
The first notion (probabilistic algorithms who are correct with high probability for all inputs) is indeed stronger. Distributional complexity relates to randomized (worst case) complexity via Yao's principle, which states that the worst case expected running time of any randomized algorithm, is no better than the expected running time of the best deterministic algorithm for any input distribution. 

You can find an exact formulation in this cstheory question, where both the Las Vegas (the randomized algorithm should always output the correct answer) and the Monte Carlo case (similar to your formulation) are mentioned.

As for your question, it is not true that an algorithm which outputs the correct answer with high probability when the input is drawn according to the uniform distribution, returns the correct answer with high probability for all inputs. Consider an algorithm which answers correctly for all inputs not of the form $0^n$. For an input drawn from the uniform distribution, the probability of getting a correct answer is $1-2^{-n}$, however for inputs of the form $0^n$ you never get the correct answer.