Yes, that is right.

As far as I can tell that mouse has an integrated battery, not easily replaceable

Yes there are but I am specifically searching for one wjth a detachable cable for charging.

Thanks for the suggestion, the battery life seems nice but I am looking for that replaceable battery.

The reasoning of a mouse looking disgusting does not really count for me, I try to keep my things clean. My current mouse is about 5 years old and still looks perfectly fine. Just its battery is giving up.