I recently discovered that the "LCD Mac" text rendering (anti-aliasing) option is the closest as type would render in any web browser.
My problem was that I didn't want to waste time by manually updating all the text layers one by one in my designs.

So, in a nutshell, here's how to update all at once :

• Step 1 : Deploy all the layers of your artboard (or document) by clicking the down arrow next to the artboard name while maintaining Option (ALT)
  
• Step 2 : Select all your layers by using Shift + Click at the top and bottom of the layers list.
  
• Step 3 : While not deselecting anything, select the Type tool (T)
  
• Step 4 : Change the font-rendering option to what you want
  

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Quick video tutorial