Your bucket name must match the CNAME (Or Alias) record you point to the bucket.
The bucket name must be the same as the CNAME
http://docs.aws.amazon.com/AmazonS3/latest/dev/VirtualHosting.html
It is not possible to rename a bucket. In the other answer you referenced, I suspect this was a casual use of the word "rename," which actually was intended to mean "create a bucket with a different name," because:
After you create a bucket, you cannot change its name.
http://docs.aws.amazon.com/AmazonS3/latest/gsg/CreatingABucket.html
So, to point a DNS entry from one of your domains directly at a bucket, you will need a bucket with a matching name.
There is one alternative, however, which does not require you to change the name of the bucket, which might be a good idea anyway: you can use the CloudFront CDN in front of your bucket.
If you create a CloudFront distribution, you can tell CloudFront which DNS names you intend to point to it, and then configure CloudFront to serve the requests from a bucket you specify -- and in this case, the name of the bucket no longer matters, because CloudFront handles the hostname-to-bucket mapping automatically.
This further gives you the potential performance enhancement of caching your image assets closer to frequent visitors of your site, and allows you to use SSL with your subdomain as the hostname (using a free SSL cert from Amazon Certificate Manager) which is only possible with CloudFront+S3, not S3 alone.