Lutron Caseta smart dimmer with Pico remote in 4-way switch set up
The pictures that you link to do not provide enough data to be able to give you specific guidance. Basically all that can be seen from those pictures is which switches are 3-way and which are 4-way. Oh and in addition where crappy back-stab connections are used. Additional pictures would clearly show the back inside of the electrical boxes so that the incoming and outgoing cables in each location could be checked to visualize the way the circuits interconnect between the four locations.

I really recommend that you install the actual dimmer at the location of where one of the end point 3-way switches are located. Ideally you will select the location where there is a neutral wire located so that you can use the Caseta dimmer that has the neutral connection. (I know that there are Caseta models that will work without the neutral connection but with the use of the neutral gives you more freedom on the types of lights that you can control). 

From  your pictures I can infer that your four way switches are using both black and white colored wires to act as travellers. This is a strong indication that there is not a neutral connection run through the 4-way switch box locations. If a neutral was present it would be far more likely that the wire pairs connected to the 4-way switches would be black and red (or some other color that is not white). 

The above observation is the reason that you want to study the 3-way switch locations carefully to see which location has a neutral present. If the power feed is coming to this switching circuit from one of the electrical boxes where the lights are located it is possible that neither 3-way switch box will have a neutral. 

On the other hand if one of your switch locations is in an electrical box that already has two or three switch units then there is a high degree of possibility that this electrical box has the power and neutral feed coming into that box.

You should not really have to care what location that you place the actual smart switch from a usability perspective. Any of the other locations will have the Pico remotes and once installed and paired with the smart switch they provide the same level of functionality from any one of your current four switch locations. 

The Caseta installation instructions will show how they recommend that you bundle the traveler connections in each electrical box to allow bypass across the two 4-way switch locations. I suggest reading through this multiple times to get a good understanding of how this will work. Also if you do not fully understand how conventional 3-way and 4-way switches work in a multi-switch arrangement I would also strongly recommend spending some time researching that subject. It will give you a much better basis to understand the Caseta smart switch installation instructions.