In which I bring in the expert
Several people have noticed that the postcard of The Times Building in the earlier entry doesn't show the eagle. So I queried Carolyn Strickler, a friend and former head of The Times History Center.
She says the post-bombing structure was an entirely new building, not a reconstruction. According to the clips, this structure (the third Times Building) opened in 1912 on the second anniversary of the bombing and the eagle was placed on the parapets overlooking the northeast corner of 1st Street and Broadway. The eagle was taken down and re-gilded before being installed on the current Times Building on June 27, 1935.
Here's the eagle being hoisted into position on July 18, 1912:
The eagle is visible in this photo of the building taken in August 1912:
Conclusion: The postcard publisher removed the eagle.
And here's the eagle in 1935:
I also found this poem written shortly after the bombing: