It was a simplification.. we meaning the voters not electors. You vote for the President of your choice and you assume his VP choice by doing so. The electoral college i.e. elect the President and VP.
Neither the Constitution nor the Electoral College prevents president from picking a VP from the same state. Electors can’t vote for both a President and VP from their same state.
Using the Bush / Cheney as an example, Cheney changed residence so the electors in Texas could also vote for him otherwise he’d have been in Texas and would have lost 32 electoral votes and we’d have had Gore and Lieberman.
The Twelfth Amendment was designed to avoid a repetition of the events of 1800 by having the electors vote separately for President and Vice President, with each elector casting one vote for each office. The Constitution’s original system at times could result, as it did in the election of 1796, in the selection of a President and Vice President with different political alignments, while the Twelfth Amendment simplified the process for selecting a President and Vice President from the same political party. The Supreme Court has thus stated that the Amendment “both acknowledged and facilitated the Electoral College’s emergence as a mechanism not for deliberation but for party-line voting.”
Since the Twelfth Amendment was ratified, Congress and the states have made other changes to presidential elections. Following the disputed election of 1876, Congress enacted a statute providing that if a state’s vote is not certified by the governor under seal, it shall not be counted unless both Houses of Congress concur. In addition, in 1933, the Twentieth Amendment superseded some provisions of the Twelfth Amendment.
Additionally:
No, that's not how Amendment XII works. Under the scenario you lay out here, had Cheney remained an "inhabitant" of Texas, along with Bush, then Bush would have still ended up with the 271 EVs he did receive (following the brouhaha in Florida, that is), while Cheney would have ended up with only 239 (i.e., the 32 Texas Electors could not have voted for both Bush and Cheney, and, presumably, they would have cast their votes for Bush). Lieberman would have still ended up with 266 EVs.
But even though Lieberman would have had more EVs than Cheney, Lieberman still would have fallen short of the 270 EVs he would have needed to be elected Vice President. In that case, the selection of the Vice President would have devolved to the U.S. Senate, which would have then voted for which of the two men would be chosen. Per Amendment XII, the Senate would have been restricted to voting for either Cheney or Lieberman.
Funny thing is, though, following the 2000 election, when Congress convened on January 3, 2001, the Senate was split 50-50. And with no Vice President having yet been elected, there would have been nobody to break the tie had the vote ended 50-50! That would have been an entertaining scenario to see play out.