To this Minnesotan in the middle of winter, pretty much what it says.
A "slick" surface is smooth, and will tend to be slippery (though a slick smooth floor is not slippery to rubber-soled shoes, for example). Ice is usually, but not always, "slick".
A "slippery" surface is one which is apt to result in slipping (and possibly falling, if walking across it). The slipperiness could be due to ice, oil, loose gravel, etc.
(Though admittedly the terms are often interchanged with little regard, especially by radio announcers looking for a different way to describe road conditions.)
The differences between black, glare and glaze ice on paved surfaces are colloquial, regional and attributive. The terms are often used interchangably, and the sense of each overlaps with the others. So, without accounting for regionalism and localism, the distinctions I know from personal experience with the use of the terms in cold climates are these:
black ice on paved surfaces may or may not appear black. It is effectively invisible while traveling, even at pedestrian speeds, but the term is applied when water freezes on a paved surface in such a way as to compromise traction. I first encountered the term black ice used with this sense in Washington State (> 1990). In my prior experience with the term black ice in Minnesota (1970-1990), it was applied exclusively to lake ice frozen without many air bubbles. In that case, the ice appears black. With that reference, such ice was highly prized by ice skaters and ice boaters, for its smooth surface and transparency, and by others for its outstanding scenic properties.
glare and glaze ice on paved surfaces, unlike black ice, reflects light, as suggested by the names, and so is sometimes visible. The two terms are used interchangably, but my observations indicate that glare is sometimes used to distinguish a thinner ice layer from the thicker layer formed by glaze ice. These observations are supported by my observations of the use of glaze ice in, for example, Oklahoma (< 1970), to describe the coating of surfaces and structures, including the twigs and branches of trees and shrubs, etc., with a layer of ice more than one quarter inch thick. While glare ice might be used with the same reference, the use of glaze as the descriptor was more common.
As mentioned, with reference to ice on pavement, black, glare and glaze are often used interchangably by individual speakers in all regions where I've encountered the terms. With reference to ice on surfaces other than pavement, the choice of term is more likely to be specific to the most notable attribute of the ice, that is, whether it appears black, reflects glare, or glazes surfaces and structures.
Best Answer
I'm British and your definition 1 of wallet is what I understand it to mean. Wallet is the only word I would use to describe such an item. It would be rare for a man to carry both. Either his wallet has a coin compartment (zipped) or he carries coins loose in his pocket. In my case, both.
A purse, in my British English, is the woman's equivalent. This is often slightly larger (too big for a pocket) and carried in a handbag, but even if it's pocket sized it would still be called a purse.
The typical usage is so gendered that the same small black leather item could easily be described as "her purse" or "his wallet". "His purse" wouldn't be used for a physical item; it sounds like an item from centuries ago, or a figurative use. "Her wallet" would also seem uncommon.
A coin purse or change purse looks like what you get for a google search on "coin purse" (no image inline as the variety is what I'm trying to show).
Your second definition of wallet (Brit: a small bag) is dated as a standalone word; travel wallet is sometimes used (mainly by manufacturers, but also if carried by someone who's also carrying something like a handbag). It may be described as a man-bag if carried by a man.