Star canes can look like many others. The example you refer to from Hollister are separate star canes, whereas in your weight the star is a central section within a thick blue surround. It is not the same type of cane. Also, it is unfortunate that earlier books, such as Hollister's from 1969, show examples of weights as "Whitefriars", which are now known to be later (1920s - 1950s) "Old English". All three images on page 171 of Hollister are examples of this error in identification.
The description in Flemming & Pommerencke, page 30, is sadly not very useful. Many people these days refer to the typical Whitefriars base finish as a "button" - which is better but still not too meaningful until an actual example has been seen.
Derek showed a photo of a Whitefriars base in another message - see here
. There is a definite concave and polished central section with a distinct indented ridge around the central area leaving a thick, rounded outer rim.
Your weight has a flat basal rim around the outer edge and the whole interior is a shallow concave area. There is no separate area within a ridged rim as with the Whitefriars finish.
Derek's comment about a cane being "pushed through" might be correct, but is not what I was suggesting. If the blue colour is simply a thin outer layer over a white core (white is often used as a base upon which colour is applied) then it may be the white core that is showing through. However, using photos, it is not easy to say what has actually happened.
Your weight, with its "basal rim finish" is interesting as it suggests an older piece but its identification may take a while. I do not recall another like like it in any of my books or for real.