What is the difference between red, white and dry vermouth?

No, colour and sugar content are unrelated; the sugar class appears as a separate designation on the label.

Regulation (EU) 251/2014 has five sugar designations: extra dry, dry, medium dry, medium sweet and sweet. They are tied to grams of sugar per liter and not to the color of the drink. In the trade many red vermouths are sweet and many white ones dry, but sweet whites and drier reds exist. So look at the designation, not at the color.

Dry vermouth is a legal designation for vermouth with less than 50 grams of sugar per litre.

Below that lies extra dry, with less than 30 grams per litre. Above 50 grams semi-dry begins, above 90 grams semi-sweet, and from 130 grams a vermouth is called sweet. The designation says nothing about colour and nothing about the herbs, so two dry vermouths can lie far apart in aroma and bitterness.

Vermouth is flavoured wine to which alcohol has been added and whose flavour is determined by substances from Artemisia species.

That plant group is called wormwood. Without wormwood a drink may not bear the name vermouth, however many other herbs go into it. A flavoured wine furthermore consists of at least 75 percent vine products and has 14.5 to 22 percent alcohol by volume.

On labels there are often Italian or French names: rosso for red, bianco for white, rosato for rosé and dry for the dry style.

Those names are trade names and not legal categories. The legal information is next to them: the designation vermouth or aromatised wine, the alcohol content and, if the producer uses it, the sugar designation. If the bottle only says rosso, you still know nothing about the sugar content.

Red vermouth usually goes into cocktails with brown spirits, white and dry vermouth into the classic martini and into mixes with tonic.

A negroni and an americano call for red vermouth, a dry martini for dry. Serving it neat is also fine: cold, with ice and a twist of citrus. Vermouth is wine with herbs, so you pour it in smaller measures than an ordinary glass of wine. Alcoholic drinks are only served to people aged 18 or over.