This is the wildest of guesses, but since no-one else seems to be in a hurry to answer this question then I might as well share my thoughts with you.

I did a web-search for winter/metal-tipped cane/walking-stick, etc. and came up with nothing that would answer your question.

Referring to the information you supply in your question, a "walker" is possibly a "walking-frame". (Maybe you knew that already.) I.e. something with four legs and a kind of handle bar to help elderly people with mobility problems to walk about unaided in their environment.

So, if the "winter-tipped cane" has a similar function it could refer to the kind of three-pronged walking-stick that I have sometimes seen used by elderly and incapacitated people.

If this is so, why it should be called "winter-tipped" is a matter for speculation. Maybe it was something invented to help people in wintery conditions. E.g. on icy roads or through slushy fields.

Who knows?

Whatever, good luck in your further search for an explanation of this term.

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Note added at 2002-08-16 06:22:52 (GMT)
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At the same web-site mentioned by the answerer below [Jack Doughty] you will find pictures of FOUR-pronged walking-sticks. They are described as follows:-

This revolutionary new cane base allows almost any cane to become self-standing on a variety of flat surfaces. Bio-medically engineered to give greater lateral support, the AbleTripodCane™ base features a flexible traction design that cushions the impact of everyday cane use through hands, wrists and elbows. The Base is symmetrically designed for both left and right hand use and is recommended by Physicians for its lightweight easy to use design.

The AbleTripodCane™ is currently being used at major University Hospital Rehabilitation Centers as an alternative to heavier, awkward Quad-Canes that are especially cumbersome on stairs. Perfect for use on soft surfaces such as grass, sand and earth.

http://www.walkingcanedepot.com/tripod

You can get canes (walking-sticks) with large rubber pads which can be fitted over the tip. They come in a variety of sizes, you would have to take the cane along with you when buying one to make sure it fitted tightly. This makes it less likely that the cane will slip and let you down when you put your weight on it.
The reference given describes and also illustrates these tips.