I know that there is a public domain (reshut ha-rabbim) and a private domain (reshut ha-yahid), but are there other types of reshuyoth mentioned in the halakhah of carrying on Shabbat?
Definition of the 4 domains
There are 4 categories of domains on Shabbat: the Reshut HaRabim (public domain), Reshut HaYachid (private domain), intermediate area (Karmelit), and exempt area (Makom Patur).
A Reshut HaYachid is defined as an area of at least 4 Tefachim by 4 Tefachim surrounded by walls of at least 10 Tefachim. This includes a pit which is 4x4 Tefachim with walls of 10 Tefachim.
A Reshut HaRabim is defined as a street which is 16 amot wide. Some say that if there's not 600,000 people passing through it daily it's not a Reshut HaRabim.
A Karmelit is defined as an area of 4 Techaim by 4 Tefachim with walls which are less than 10 Tefachim in height. It also includes areas where the public doesn't walk and doesn't have proper fencing such as fields, or rivers which are 10 Tefachim deep.
A Makom Patur is a place which is less than 4 Tefachim by 4 Tefachim but is of at least 3 Tefachim in height. They are only a Makom Patur when placed in a Reshut HaRabim, but in a Karmelit they are considered like a Karmelit.