I know the "2nd place" offers are always just rumors, but if you can find some examples of guys taking a meaningful amount of less money to sign for one team over another, then I'd love to hear it. They all talk (conveniently after they have signed) about how all the outside factors swayed them to this team over another, but it's always about the dollars. Like Heyward, where his comments were all about "young core" and "large window" but failed to mention the front loaded contract and additional opt out that made it a better deal.
That article mentions Darvish, Cishek and Chatwood, all of whom took their best offer on the table this off season, according to reports.
The only one I can think of off the top of my head that comes close is Justin Turner before last season. He had some offers reportedly for $75MM/ 4 yrs and took whatever the best Dodgers offer was, just because he had no interest in leaving his hometown. Those type of "hometown discounts" almost exclusively come in extensions, where the team is taking on some risk in exchange for more years of control on the player.
It's not really comparable to a salary capped (and individual contract capped) league like the NBA. Because if the Bulls could offer Kevin Durant $60MM and the Warriors only offered $20MM, I guarantee he wouldn't be on the Warriors no matter how inept GarPax is.