I have a Konica Minolta 'Dimage Z10', and am trying to calibrate the 'White Balance' for indoor work.
The instructions suggest that the camera should be focused on a sheet of white paper (with the whole area in the frame), and with Aperture Priority selected, the 'Custom Set' be used to register the 'white' value of the paper. I then photograph my subject (a single model or a diorama) against a similar white background. Although the result is successful in taking away the yellow cast from the artificial light, when the images are viewed on my computer, the background invariably appears grey.
Am I missing something important?

These WB corrections will work if you make sure the background receives exactly the same kind of light source as the one you use on you white reference. WB correction will never work if several light sources are fighting against each other in your photo.
Second possible issue, your white reference might be not really white... I purchased few weeks ago a set of black, white and 18% grey cards used in photography for WB. I preferred the A4 size to make sure I cover the whole picture frame, as in my camera the whole picture is used for the correction and if you dont do that the surroundings might add some unwanted cast. These color card sets are quite cheap.
Additionally maybe you see the photos in a non color calibrated display...
WB is almost black magic...
Kind regards
Nacho

Apart from what Nacho says, ensure you are using white light bulbs (or leds). Try also to measure the white with the real background, not a different paper. And check if the photos on other screens look the same...

IF you need manual White Balance do it on the spot. That is WHEN and WHERE you take the pictures. And as astursimmer suggested use a Whitecard or greycard like https://www.youtube.com/watch?v=zya_8TBvctY shown here (Video, Foto - all made in Japan)

And yes, ultimately you want a calibrated monitor as well using ie. the Datacolor Spyder for calibration. You can do some calibration with alternate techniques but from experience they will not match the (not so cheap) Spyder or similar products (I use the Spyder 5 Pro on my Windows system)

IF your camera supports a "digital negative" format (often called RAW) AND you have stuff like Lightroom/Capture1/Darktable etc. in use you can adjust white balance in post processing as well. Before buying or installing the software check if your rather old compact camera is supported.

I generally do final white balance corrections on the computer rather than trying to get the camera perfect.

For modelling I do this by making sure there is a white piece of card/paper in the image somewhere under the same light conditions. I can then select that object as my white balance reference, and then crop it out of the final image. Not all image software supports white balance correction though.

If there is a decent sized white segment on the model already then I can use that as the reference instead and reduce the need to plan for cropping the image.

I do try and use decent lighting and background conditions to get the best image I can to begin with, so for that I use a light box.