Nathan,
  You are entering the wonderful and mysterious world of Jaguars.  You will
not find two Jags from any year identical to each other. (Much less to other
years).  Remember that your '88 is only like the '87UK version and the '89
XJ40.  Starting in 1990, the cars are VERY different!!!
   Your '88 XJ40 originally ran the power steering on ATF Dextron II.
Dextron II is no longer made so Dextron III is the replacement.  You have a
separate reservoir on the right side of the car that is black plastic and
says "Power Steering Fluid" and has a lid that has reverse threads. Right is
loosen.
   As Jag "progressed", they blended the power steering with the Green
fluid for the brake assist and the SLS in the 1990 models. That is most
likely why the PO told you to use the "Green Blood" in the Power steering.
   Then you have the guys like me with teenagers that drive these cars.
When the lid says "Power Steering Fluid" on it, my kids an not going to put
in ATF -- they will put in Power steering fluid.  So I converted my fluid
over to the clear power steering fluid.
     All three of these will work and work fine.  They are different
chemical composition, but have essentially the same characteristics for the
power steering needs. The green blood will seep through the rubber hoses and
leak (weep) the quickest.
       The question is cost.  The "HSMO" Green fluid is about  $10 per
litre and a bit difficult to find.  The Power steering and ATF fluids are
cheap and can be bought at the local store anywhere.
       Here is what I would do if I were in your position. Leave the HSMO
(Mineral Oil) in your power steering rack.  It works just fine. Go to the
local auto parts store and buy a plastic bottle of BARR'S stop leak for
power steering for HONDAS.  It is the same green color as your Power
steering fluid. Keep a bottle of this in the trunk so if the fluid levels
start to drop, you can use this as stop leak and lubricant.  Eventually,
your rack will go.  Mine went at  130,000 miles.  When it does, you will
have to replace it and the fluids. Simply convert the fluids at that time
back to the Dextron and wire a little tag on the reservoir that it uses
Dextron III so no one blends the fluids in the future.

One of the other reasons I went with Power steering fluid was so I could
tell what was leaking onto the garage floor in a more ready fashion.  My
fear was I would see red ATF fluid and start to tear the transmission apart
when it was only a low pressure $5 hose from the power steering.   Personal
opinion

Cheers
Webserve