Cartridge filters don't get multi-ports. They are directly plumbed from the pump and to the heater. Someone will use the multi-port. It has a back wash setting. They will cause the cartridges to tear themselves apart. I've seen it happen before.
But since the tanks for Pentair's cartridge and DE filters is the same except for the labels, it may be a DE filter. The only way to know is to open it and look.
WRT to the offer, if the home isn't already below market value, I would subtract an additional 15K for work needed from an offer that assumed the pool needed no work beyond cleaning. That would cover plaster, tile, spa wall repairs if they are as needed as I suspect, and maybe coping too, if it is loose in places.
It's a good bet that they won't see it that way
I don't know what other work the house needs or real estate markets. Some put a sizable chunk in escrow for such things. If your offer is accepted, have the pool checked in addition to the usual inspections.
Tile doesn't just pop off. Something is happening behind it usually. That something is usually crumbling cement.
Scott