I'm going to restrict my observations to the last video, since I can't tell the scale and initial conditions in the first videos, although I suspect that everything I'm about to say also applies to the first videos.
Our relevant forces are: gravity, cohesion (attraction between similar molecules), adhesion (attraction between water molecules and dissimilar molecules), and surface tension (tension at the air-water interface caused by the bulk force imbalance between the cohesive force of water and the lack of an adhesive force between water and air which tends the liquid towards spherical shape).
What you are observing in the video is convection in a container. The surface tension of the air-water interface functions as the boundary of a flexible container, which is glued to the solid barrier by the adhesive force of the water-barrier interface.
Consider what would happen if we poured powdered water slowly into the center of a half-filled glass of water.
The powdered water will process down the center of the glass, propelled by gravity and momentum, displacing the other water, which must rise up the sides. As the powdered water nears the bottom, the leading edge is already smearing out, as the cohesion of the displaced water pulls it along into the convection rising up the sides of the glass. When the powdered water strikes the bottom, it all must join the convection cell and begin to rise up the sides of the glass. As a particular section of water reaches the top of the water in the glass, it is moving too slowly to overcome gravity and cohesion forces, so it doesn't continue upwards. Rather, it rolls along the surface and back down the center.
Now, we are left with a slowly rolling convection cell in the glass in which the water at the center is flowing down, and the water at the sides is flowing up, even after we stop adding water. Eventually if left alone, entropy will prevail and the convection cell will disappear, leaving random Brownian motion.
Of course, there's nothing special about the powder - if we did the same thing with clear water, the same effect would be happening. The powder just lets us see it.
The small splash of water in the video is contained on one side by the rigid barrier - and on the other side by the flexible boundary of the air-water interface, created by surface tension.
When the barrier is tilted, two inseparable things happen to the water splash. The water begins to flow in the direction of the tilt, pulled by gravity, and the surface bulges near the center in the direction of the flow of water, because surface tension causes it to seek a domed or spherical shape.
Water near the bottom of the splash stays stuck where it is by the adhesive force between the water and the barrier. The strength of the adhesive force decreases very quickly with distance from the interface.
The domed shape of the water means that at the edges, the water is shallow. The adhesive force keeps the bottom layer glued to the surface, and the cohesive force keeps the adjacent layers from moving too fast. At the center, the water is deep. The bottom layer feels the same adhesive force, but far above it, the water responds to the gravitational force and flows downhill. Cohesion pulls the rest of the center into a downward flow. So, like when we poured the powdered water into the glass, we have an initial condition in which the water at the sides is almost stationary while the center is flowing down.
Just like the water in the glass, our splash is bounded at the end of the flow path - not by a solid barrier, but by the surface tension of the air-water interface. The flowing water doesn't have enough energy to break the surface tension and form a new rivulet continuing down the barrier. With nowhere else to go, the downward flowing water pushes the rest of the water back up the sides of the splash.
From here, everything works exactly like it did in the glass. (Except that the convection cell in the glass is radially symmetric through the whole container, while the convection cell in the splash is bilaterally symmetric locally only.) A convection cell forms, or starts to. (Or several cells form, because of the irregular shape of the splash "container"). Water flows down the center and up the sides, then back down the center (if there's enough energy and a sufficiently regular container for a full convection cell to form), until eventually entropy prevails and the convection cell disappears.
In the last video, the convection process gets set up because you tip the barrier.
In the first videos, I surmise that the convection process got set up by whatever process deposited the water there in the first place. Because adhesion and surface tension affect moving water just as much as stationary water, the rivulets were moving very slowly if at all at the edges and quickly at the center. When the source of the flow ran out and the rivulet stopped moving down the sidewalk, the end of the rivulet started acting like the bottom of our glass.
