It helps to think “coil” when you hear the word “phase”. A common misconnection, even among electricians, is that the wires coming from a transformer are the phases. They are not. They are phase legs. A coil makes a phase.
Every coil has two legs extending from it. It may also have one or more taps. Single-phase has one coil, two-phase has two coils, three-phase has three coils, etc. The transformers and the connections tell you what the service type is.
A proper two-phase five-wire service will have all five conductors landed on a single disconnect. There can be multiple disconnects but all five conductors have to enter the building together into the same enclosure. From what I can see in your photo of the transformer connections, it appears to be a two-phase connection. However, your photos from the panels do not show a single entry point.
I developed a very effective classroom exercise that illustrates isolation transformer connections. Once you see this classroom demonstration, it all becomes very clear almost instantly. It is difficult for most people to visualize real transformer connections from schematic diagrams. I don’t have any photos of two-phase five-wire transformer connections but I do have a schematic.