how to update Safari from 6.1.6 to 9.1.2?

You would have to upgrade your Mac to the newest OS X El Capitan.

Upgrade to OS X El Capitan - Apple Support

Your Mac is running v10.7.5. Safari cannot be updated beyond version 6.1.6 on a Mac running Lion v10.7.

It's not a scam. The only way to update Safari is to upgrade OS X meaning El Capitan.

El Capitan is free and available at the Mac App Store. El Capitan requires about 9GB's of available storage space.

Hi Ganlareej, yes, the pop-up you got was a scam, never listen to a third party site that asks you to call or download anything. The only way you can upgrade your Safari is to upgrade your system. The latest version of Safari 9.1.2 is available if you upgrade to OS 10.9 or greater.

yes, the pop-up you got was a scam

No, it wasn’t.

never listen to a third party site that asks you to call or download anything

It’s not updating software in response to a website prompt that’s the issue. It’s downloading the software from an untrusted source.

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Live and learn.

Back up your data,check your applications for compatibility, and download El Capitan from the Mac App Store. If you have access to Mavericks or Yosemite, you can upgrade to them instead.

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You upgrade the OS as described or switch browsers, but don’t delete anything.

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If you upgrade your OS, it will replace the files of the current OS, so you aren't leaving files there that take up space.

It is always a bit embarrassing to get a Solved for being no help at all. Carolyn and Niel had the answers that helped you.