A Stateful address assignment involves someone keeping track of the State. Which is to say, some system exists that provides a log that certain IP addresses were assigned to certain MAC addresses. DHCP / DHCPv6 keeps truck of such information.
A Stateless address assignment does not keep track of what has or hasn't been assigned. It simply determines what address it should use on a particular network (to include possibly picking it at random after determining the prefix) and uses a system like Duplicate Address Detection (DAD) to see if it inadvertently selected an address already in use.
In a Stateless assignment scenario, there is no server or device which keeps a running log of what IP addresses have been assigned and what IP addresses are remaining, available to be assigned. The mentality in a Stateless assignment scenario is: Pick an IP address. If it happens to be in use, pick another one.
TLDR: Stateful requires a DHCP server to have been configured to hand out addresses. While Stateless address assignment simply requires the Router/Default-Gateway to understand and implement RFC 4862
Your assertion is essentially correct.