I don't understand how you would be disrespectful to your employer by taking a sick day and doing an errand or using the computer, no matter what Miss Manners says.
I can understand the idea that you shouldn't take sick days unless you're actually not fit to work in some way or other. I personally have no problem with people who use sick days solely for errands or other non-sickness related needs, as long as it doesn't get out of hand. Especially if it's set up in advance and coverage is arranged. But at best that's a slippery slope, and anyone who says sick time should only be used for sickness, period, isn't going at all overboard in my opinion.
But as you say, if you are sick, that's the end of it. If you stay home because you have a migraine, or a lousy stomach, or whatever, that's a legitimate reason to not go into work, and as far as I'm concerned work should have no interest whatsoever in what else you do with your day, within some very broad limits. The idea that someone who takes a sick day should stay off the computer for fear of someone from the office deciding they're not really sick is absolutely ludicrous, in my opinion. You don't have to be immobile in bed for the entire day to qualify for a sick day, you just have to be in a state where going to work would make you very uncomfortable and/or put others at risk of catching whatever you might happen to have. "I've got a cold and feel like crap" is plenty good reason to call in sick and go back to bed. If later on you wake up and noodle around the internet, who cares? Most for-fun internet activities don't require nearly as much attention and thought as work activities, for one thing, and being able to concentrate on a computer in the comfort of your own home while sick is much, much easier than doing the same basic thing at work.
And the idea of being tattled on for daring to leave your home to go and buy medicine leaves me absolutely appalled. As far as I'm concerned the only person who should have been fired in that mess is the person who actually thought it was appropriate to fire the sick woman. What if someone had seen her driving herself to the doctor? Would that also be proof that she's obviously well enough to work? Bah. As you said, sick leave is a benefit given to workers, it's not a privilege that needs to be defended and justified every time you dare to make use of it.