The union contract directs how the airline and union work together. It may define how the airline is permitted to formally make an offer, but it likely also defines how the union can formally respond (doubt it's via the NYT). We could argue that both sides are "wrong" in this trivial administrative matter (how they communicate).
Most citizens don't work for AA and those that are party to this negotiation wouldn't be permitted to discuss it here. Therefore, this is entirely an issue of public relations and reputation (for AA and the APA). The public saw AA make an offer. The public saw the union reject the offer for reasons other than the offer's merit. The public did not see the union respond in a way that shows they desired to resolve an issue in the interest of their customers. The following press release from AA management thanks everyone - including the union.
Regardless of what the contract says and what the parties say now: AA management, from the public vantage point, took the high road while the APA did not.
PS: Look back at the Toyota unintended acceleration problems around 2004. Toyota was technically correct in their responses to the public, but still lost their reputation and many many $millions because they failed to properly manage the story. Being contractually correct didn't help.