looks like you were hitting gold with the second shovel you took - great car and of course the best model of all series Jags;-) Congratulations!
The “reserve tank” isn’t a reserve tank, but a second tank with a second pump and a second fuel level sender. The guage is the only part shared - wouldn’t it be cool to have two guages anyway …
With the PO telling funny things and you obviously not fully aware of the setup it might be helpful to start at square 1: Upon removing the spare wheel cover and the spare wheel you can undo the rear part of the plywood cover - it is about seven screws I think (note where they were located as they are not identical). Then have a look at what you find and compare it with the original documentation.
Earlier and later cars have two fuel pumps fitted and it looks pretty much like this http://www.jag-lovers.org/snaps/snap_view.php3?id=1426971017.
Maybe the PO’s explanation and your assumption of a switched reserve tank got you confused and everything is just as it was designed: the engine uses one tank that is selected with the button you mentioned. In normal position (green light with instruments illumination on) it selects the right tank and the fuel guage shows the fuel level of the right tank. In depressed position (red triangle light pointing left ) it is the left tank. If everything works as it is supposed to the switch operates a valve in the trunk, selects the fuel pumps and tells the fuel guage which fuel sender to use.
This complicated setup was frequently “simplified”, either because owners only wanted to fill up both tanks through one filler cap or because defunct fuel pumps, valves or senders seemed too costly to replace. When defects were limited to one tank it is quite common to find part-disabled fuel systems.
So if there is a defect in the first place, you might check whether the tanks and the fuel system in the boot are tight at all. Fill both tanks completely and you will smell whether there are any leaks. Take care though, fill up slowly and keep a keen eye one the offending tank. And to be honest, it isn’t easy to really get everything tight even with good tanks. Pinholes at the tanks or along the metal tubes are a PITA to find and fix.
If everything is good so far, chances are the fault is limited to the switch in the dashboard. These tend to lose function with lack of use. You can try to exercise the switch to restore function. You can pull it by carefully prising out the clock, reaching in and pressing the switch panel out towards the cabin. Some cleaning and lubrication often does wonders …
Let us hear how you get along with it!
75 XJ6L 4.2 auto (UK spec)