The "first" stimulus (not really first) had extremely little content that even conforms to your explanation of good stimulus spending. Thus, by Keynesian economic standards, it should have failed. It has failed so far and I see no reason under either Keynesian economics or my preferred economics to think that will change.
What will a "second" stimulus change? Will our government do an actual Keynesian stimulus, or will it pay lip service to an ideology it believes tells it to do what it wants to do anyway (always dangerous!) and just spend, spend, spend? And if your answer is the former, I ask you, on what evidence do you base this belief since all evidence seems to point against it?
Explanations of how Keynesian stimulus works are pretty irrelevant in a world where we aren't actually using them. People who think the government should cut back may be wrong in the sense that it is not optimal, but it may still be the best course that we can actually bash Congress into following. It isn't a chamber where "nuance" does very well.
I have read vast swathes of the "first" stimulus bill. I invite you to read it and come to your own conclusions about exactly how much of it is actually Keynesian. I didn't find much that was unambiguously Keynesian, much that was unambiguously not, and a lot of other things that are basically slush funds that seem pretty unlikely to go to Keynesian things.