The pictures don't help me...maybe a plumber.
If the house was plumbed for a recirculation loop, at least one fixture would have a T on the hot pipe so that the hot water could get back to the WH.
Where is the WH in relation to those pipes in the picture?
The simplest loop would be one that uses gravity. You do need a check valve to prevent water from the bottom of the WH (where it is likely cold after some use) to be pulled out of the hot line at the point of use (i.e., two parallel paths...some will try to flow both ways).
To keep the energy costs down, when there's a pump involved, you typically need something to slow or stop the recirculation once the end point of use gets hot. On some systems, they turn off the pump, on others, there's a valve that closes.