Do light cones "tilt" towards black holes? In some diagrams, light cones becomes "thinner" near black holes. Meaning the light has trouble moving nearer or further away. As in this picture. I presume this corresponds to someone observing that things never reach the event horizon.
Whereas some diagrams have the light cones tilting towards the black hole in a time asymmetric manner. As in this picture. And also things moving inside the event horizon.
Now seeing as gravity is time symmetric. (Throwing a ball and catching it would look the same backwards as forwards). I would expect the first diagram to be correct. 
I also have a feeling that you will tell me that somehow both diagrams are correct from different perspectives.
I can understand the time symmetric diagram as this corresponds to the Schwarzschield solution.
Can anyone tell me if the second diagram with tilted time asymmetric lightcones is also correct, and if so how? (And what is the relation between the two diagrams?) How can one have a time asymmetric diagram of gravity if our experience of gravity is time symmetric?
Could we have a time symmetric diagram in which things can come out of a black hole and fall back in again?
 A: There is no "correct" shape of the light cone, because a drawing of the light cones is always done in some coordinate system, and the result will look different in different coordinate systems. Away from the event horizon, I can use a coordinate change to make the light cones point inward, outward, or even upside down.
Only one thing is for sure: as long as the coordinate system is not singular at the horizon, and you are consistent with your time orientation, future light cones exactly at the event horizon must point solely into the black hole. As you noted, Schwarzschild coordinates don't obey this, but that's just because they're singular at the event horizon. In Schwarzschild coordinates it takes an infinite amount of coordinate time to enter the black hole, which is why the light cones close up. 

Could we have a time symmetric diagram in which things can come out of a black hole and fall back in again?

Kind of, in the Kruskal extension of the Schwarzschild solution. In this solution there is a white hole that you can only exit in the past, and a black hole that you can only enter in the future. In some sense this is the analogue of throwing a ball up and having it come back down.
However, this is not a physical solution because of thermodynamics. Even though white holes and black holes are related by time reversal, only black holes can exist in our world. So, like most fancy stuff in general relativity, it's a moot point, without any observational consequences.
