First of all, thank you so much for helping me! :) I'm really glad we could get it to work!
Ok, so what I did is basically followed the steps that @tmethei explained in his thread, but I substituted in the updates I would have to use for my computer (because I have a Macbook air instead)
I started off by backing up my GraphicsCard.sgr and GraphicsRules.sgr files (I saved them outside of the Sims 3 folder just to make sure the copies wouldn't confuse things)
Then I went and checked the information and found this...
Vendor: Intel (0x8086)
Device ID: 0x1626
So I went into the GraphicsCard.sgr file with TextEdit and changed these lines...
card 0x0152 "Desktop IvyBridge GT1"
card 0x0156 "Mobile IvyBridge GT1"
card 0x015A "Server IvyBridge GT1"
card 0x015E "Reserved - IvyBridge GT1"
card 0x0162 "Desktop IvyBridge GT2"
card 0x0166 "Mobile IvyBridge GT2"
card 0x016A "Server IvyBridge GT2"
With this...
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
card 0x1626 "IntelHD6000"
After that was done I opened the GraphicsRules.sgr and replaced this line...
elseif (match("${cardName}", "*IvyBridge*"))
With this...
elseif (match("${cardName}", "*IntelHD6000*"))
Ok so lastly I added in the final change that you told me to do...
After the line card 0x016A "Server IvyBridge GT2" (which of course is now: card 0x1626 "IntelHD6000") hit Return to create a new line.
In that new space type in: card 0x1626 "Mobile Broadwell GT3"
And then I opened the game and it started almost immediately! Problem solved :)
