Not quite… vodka is a neutral distillate, meaning that it’s essentially ethanol and water, with as little in the way of flavor compounds/congeners as possible. US-made vodka is required by law to be only ethanol and water.
The other distillates such as tequila/mescal, brandy, whiskey and rum are white and high proof coming off the still, but they’re not distilled precisely enough (usu. intentionally) to be flavorless like vodka, and retain a significant character from the starting wash. They’re typically aged in wooden barrels, although there are clear styles that are essentially unaged (white rum and silver tequila are the most popular two).
To use an example, there are vodkas out there distilled from grapes. Ciroc, a rapper favorite, is a good example. The ONLY difference between these and brandy is the particular cuts used during distilling, and the aging. They distill more precisely to only get ethanol (pretty easy using a column still) and they don’t age it for vodka, while for brandy, they use a wider cut in order to retain more of the grapes’ character, or they may use a pot still, which by design gives you a wider cut. Then they age it in wooden barrels. But taking Ciroc vodka (distilled from grapes) and putting it in barrels wouldn’t give you anything approaching traditional brandy. Technically, Ciroc probably IS brandy, although it’s also vodka.
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As to the OP’s question, the main reason you have so few categories is that a few of them are defined so broadly that they encompass almost anything out there.
You have whiskey/whisky, which is distilled from various grains and aged in wooden barrels.
You have rum, which is distilled from some kind of sugarcane products and may be aged in wooden barrels.
You have Brandy/eau-de-vie, which is basically fermented and distilled fruit, with eau-de-vies being the straight, unaged distillate and brandies being the aged version Cognac and armagnac are specific aged sorts of grape brandies.
You have tequila/mescal, which is fermented and distilled agave, with tequila being mescal made from a specific sort of agave, in a specific region, and following specific methods.
And finally, you have vodka, which is one of the above, only distilled extremely precisely, and possibly filtered to remove all character, giving straight ethanol and water.
Pretty much anything out there is going to fit into one of those broad categories, or be pretty close to it in terms of process or flavor. For example, the Brazilians claim that Cachaça isn’t rum due to minor process differences, but in a broad sense, it is.
About the only thing I can think of that wouldn’t slot into a category neatly would be something like a sorghum syrup distillate. Would that be rum or whiskey?
