Update beyond Safari 6.1.6 or upgrade to OS X required?

Hello. I'm having issues with a Internet app I've used before and the App owner is telling me my Safari is out of date. I'm running Mac OS X, V 10.7.5 and Safari V 6.1.6. I believe Im up to date but not 100% sure. I'm assuming if this version is up to date within Lion OS X, then would my next update be Mountain Lion or Yosemite, although many seem to be having issues with Yosemite, or maybe just perceived. Any info would be appreciated. Thank you, jerry