The setup in my machines here is equal to the picture you posted, of course not all lines to the reagents on the 10port VICI valve and the additional set for the second flow cell is not shown there. But maybe the setup is slightly different between early and late machines of the HiSeq 2000/2500 product line.
The syringe pumps are used to pull the exact volume to sufficiently displace the fluid of the previous step from the flow cell. The setup with equally moved 8 syringes (one for every lane) and a binary flow splitter in the intake is used to ensure that every lane is flown through with identical volumes.
The 250µl pump is used to fill/flush the lines from the reagent chiller to the 10-port VICI pump without touching all the rest of the fluidic system. Maybe somebody can post a full sequence of pump/valve actions already extracted from the logs?
The parts are shown on the photo i posted, see the files “LPVX dispenser pump (www.theleeco.com).pdf” and “Bio-Chem Isolation Valves.pdf” in the Fluidic section of the documentation package i provided. I was happy about this pump/valve combination because i was able to reuse them for a setup with capillary electrophoresis.