The linked question is great and it addresses the issue of being viewed as a slacker, but I want to address something you brought up which isn't covered in the answers there(at least, not that I saw). You said:
I am trying to move up at work and the only advice I get from people is "come early and stay late". However, I would have to give up many healthy activities that I do around 5-6pm in order for me to do that.
The thing is, this leads to burning out. Which leads to undue stress and anxiety, and makes you less productive overall. This is actually documented in studies and I'll leave that as an exercise for you to find more information.
So "come early and stay late" leads to stress, anxiety, and less productivity just by itself. Then you remove your healthy activities from 5-6PM(or showing up late at 5:10, 5:20?) - so you're effectively taking a huge hit to your health. The question is, is it worth it? In some cases yes, in other cases no. But I don't personally see this as a successful long-term arrangement. You need to take care of your mind and body.
Some wisdom I heard a while ago goes something like this: When you interview, people ask about your work ethic, your experiences, and your drive to succeed. They don't ask how often you burned the midnight oil. The job description tells you how many years of experience you need, but it won't state hours of overtime.