To be clear, the font weight of 500 is showing up in the published site. In fact, if you look at the published site in Safari, Chrome, or Firefox, it's displaying 500. Proxima Nova 500 is Proxima Nova's "Medium" font weight. What you're seeing in Safari before publishing (inside the Webflow Editor) is Proxima Nova's Light font (300). Yes, this behavior is odd in Safari, but 500 is what you've told it to do, and in the published site, 500 is what you're getting.
Here's a side-by-side using Safari on a non-retina display. Top is the published site, bottom is the Webflow Editor.
And here's the official font sample on Adobe's Typekit website, which demonstrates that you are indeed getting the font weight of 500 on your published site:
To summarize, this does seem like a bug in Safari or Webflow which is showing the incorrect font in the Editor. I've edited the post title and category to reflect this.