Refreshingly, only a handful of comments are the expected trollish remarks saying that folks shouldn't work in entry-level jobs if they don't want to follow orders and so forth. This shows some ignorance about the culture there (as well as ignorance about how great it is to live in a country where we're free to express our opinions). Having dated a Starbucks manager, I know that they actually do value giving power and a voice to employees at all levels. This is not of course to say that working there is comparable to being a partner in a small business or anything; you're still an entry-level employee at a large company, but the company does make efforts to be decent to people.
One comment stood out to me in particular. "I'm a fat guy and founder of the largest size-positive group in the Northwest. … As for her claim that fat people will be offended by use of the term, there will always be people with a victim's chip on their shoulder looking for any excuse to place blame. As for the 13,000 people in our Seattle-based group, we KNOW we're fat!" Interestingly, the original letter did not specify that the term would be offensive exclusively to people of size, but rather, that it can bring up issues of body image for some people. I find it really intriguing that the commenter is dismissing her concerns based on the views (he assumes are universally) held by a fat-positive group. This is a group of people who have sought out a group of others for support and camaraderie in identifying proudly as people of size. These are probably some of the people with the healthiest body images in the world. Not to say that they all came by it easily, but people in a size-acceptance group are just not going to represent the average person in terms of body image.
What I think is the real issue here is not that Starbucks is using a term that can also be used to describe body sizes, which is what most of the commenters seem to be focusing on. If a hardware store were calling out orders for "skinny dowels" or something, or if an auto maker advertised a "skinny seat" that folds down to a low profile, I wouldn't think anything of it. The word itself as a descriptor of girth isn't harming anyone. Just hearing the word "skinny" to describe something other than a person isn't likely to have much of an effect on body image or disordered eating. As the original author pointed out, it can be complementary or derogatory when applied to the human physique. It's a rather neutral word without context.
The problem I have is that their particular application of the word encourages dieting behavior. The emphasis isn't on choosing products with more nutrient value and less junk in order to take better care of one's body. Their campaign is that you buy their skinny drink, in order to diet, in order to be skinny. And of course, the only reason to be skinny (rather than strong, or healthy, or energetic, or comfortable in one's body) is because society says it's the right way to be. And how does society let us know how we should be? Through large corporations and advertising in mainstream media is one way.
Starbucks is a particularly interesting example to play with, because they have quite a few nutrient-dense choices there. They've always had high quality juices and smoothies, they've had soy milk since at least the mid-'90s, and they have whole grain breads and fresh fruit and vegetables. Do people order these things? Some do, and presumably enough to meet the demand to keep them there. But where's the advertising catchphrase for ordering one's food/beverage with more whole grains, or more soy protein, or more fresh produce, or fewer empty calories, or fewer chemicals, or less processed stuff? We have language for making these requests, of course, but no universal terms like "supersize" or "diet" or "light" or, of course, "skinny." Not to mention that the menus and advertisements rarely tell us the health benefits from the various choices of breads and milks and condiments offered. This can have pros and cons though; if a restaurant doesn't have catchphrases labeling the perceived benefits of a particular meal, we don't know what the goals are of the people around us. We just might notice that the people in line at Starbucks each placed a different drink and food order. But when we throw in the catchphrases, then we keeping hearing this word "skinny," which must be a legitimate and correct way of ordering food, otherwise why would they have a word for it? And we notice how many people are eating in this way. We don't hear anyone's order getting tagged with "healthy" or "nutritious," so these things aren't entering our minds as anything on which we should place great importance. The options and catchphrases only tell us how to get our fix of either empty calories or calories-replaced-with-chemicals -- both of which are disordered eating patterns. It can be hard to ignore these messages and just live by the concept of "eating in order to consume nutrients that my body needs so I can be healthy."