Rsync creates a directory with the same name inside of destination directory

If you really want to use rsync, it sounds like you'll need some combination of --backup, --backup-dir, and --suffix. The closest I think you could get is with something like this

rsync -abv --suffix R1 --remove-source-files src/ dst/

This would do close to what you want, but it would not rename the files exactly the way you'd want. The --suffix option appends text to the end of an existing file, but it only does this for the first conflict. If you ran it again, it would just overwrite your first backup. You'd have to change that suffix value each time the command ran, which would work if you used something with a timestamp, such as this:

rsync -abv --suffix `date +%Y%m%d%k%M%S` --remove-source-files src/ dst/

I'm not sure if this is overkill for what you're after, but it should meet your requirements.

My answer, which I give from hard-earned experience, is: Don't do this. Don't try to copy a directory hierarchy that makes heavy use of hard links, such as one created using rsnapshot or rsync --link-dest or similar. It won't work on anything but small datasets. At least, not reliably. (Your mileage may vary, of course; perhaps your backup datasets are much smaller than mine were.)

The problem with using rsync --hard-links to recreate the hard-linked structure of files on the destination side is that discovering the hard-links on the source side is hard. rsync has to build a map of inodes in memory to find the hard-links, and unless your source has relatively few files, this can and will blow up. In my case, when I learned of this problem and was looking around for alternate solutions, I tried cp -a, which is also supposed to preserve the hard-link structure of files in the destination. It churned away for a long time and then finally died (with a segfault, or something like that).

My recommendation is to set aside an entire partition for your rsnapshot backup. When it fills up, bring another partition online. It is much easier to move around hard-link-heavy datasets as entire partitions, rather than as individual files.