Trying to keep your body heat from the wine is only necessary for chilled wines (whites, for the most part). For those served at room temperature (most reds), you actually want your body heat to warm the wine.
I've been to plenty of places that use 'stemless wine glasses' (about the size of a red wine glass, maybe a little larger, but no stem underneath) for reds. I've also been to places that just use mason jars.
For whites, if you're at a dinner table, it's not going to matter as much, as you can put your glass down. The problem comes when you're standing around with your glass. You can get around the problems by making sure that the wine is well chilled, and don't serve as much per glass (so they don't hold it so long that it warms up). Handled glasses can help, but I wouldn't go for a large beer mug like Cos recommended -- I'd use a smaller mug for coffee or tea.
One exception would be your bubbly wines. You want something tall and narrow for those.
None of these have the characteristic wine glass shape, with the smaller opening at the top, which is going to help concentrate the aromas, so if you have any glasses that are shaped like that, consider using those. And you don't want to fill any glass of wine more than half full, so that you have space for the vapors to collect.
Best Answer
As my answer is quite long, it was suggested to add a summary up front. Here are the main points along with a little more info.
In the US, commercial cooking wines found in the grocery store, usually on the aisle with vinegar products, contain salt and other preservatives. The main reason for this is stability, giving the products a longer shelf life after opening. Other reasons may include taxation and regulations. The same is true of cooking sake.
In the US, drinking wines that are usually cheap or of lesser quality may be considered cooking wines. Sources state that the difference between these wines and wines that one would prefer to drink is quality. Also, these wines may contain a higher amount of alcohol than those labeled as cooking wines. Due to a higher alcohol content, one may achieve more complex or deeper flavors.
Based on other answers and comments, salted wine doesn't seem to be common in many other countries.
Some wines that are considered to be cooking wines, such as Shaoxing (which is a Chinese cooking wine) seem to be quite easy to find with no salt.
As per my comments, I don't feel that there is just one definition or that one blanket statement gives a full and complete answer to this question. Thus, my quest to provide an answer that covers more ground.
Cooking wine, depending on location can mean several different things. As noted in the answer above, it is different in Germany than in the US. Even in the US, there is much conflicting information, as well as many different products considered to be cooking wine.
From Wine Folly, which is written by a certified sommelier:
From leaf article: Differences Between Cooking Wine and Drinking Wine:
As for why salt or other additives are used, taxes and regulations could be a factor, but the most important factor is that it lengthens the shelf life after the bottle has been opened. Also from leaf:
From Holland House, either the largest (or one of the largest) manufacturer of 'grocery store cooking wines' in the US:
(Emphasis mine.)
So, to sum up, there are many different products or 'wines' considered to be cooking wines. There is not just one single answer to the definition of a cooking wine. It totally depends on geographical location and which 'wine' is in question. Is it a commercial cooking wine? Is it a poor or lesser quality 'drinking wine' that is typically considered to be a cooking wine? There are too many variables to sum it up in one brief sentence.