There are a couple of details that might help here. For example, what's your "council". The fact that there is one that you are affiliated with, and it has rules/bylaws, tells me that there is some existing structure going on.
In general -- yes -- the acronym isn't all that meaningful. It's the details behind the acronym. PTOs and HSAs and PTCs are all -- in effect -- local parent organizations, typically organized around one school or one district. You could say they are all school support organizations. How formally they are organized is the difference, but that's unique to the school, not to the acronym.
Groups that are formally part of the PTA do have structural requirements. They have standard bylaws that must be followed and dues that must be collected and paid to the state and national organizations, etc.
It sounds like you are advocating for a good direction -- more systems and structure and organization. Those things typically serve a group well, no matter the acronym.
But there are definitely groups that operate more informally and still wind up doing good work. You may do better with "should -- here's some good reasons why", as opposed to "must".
I am an officer that serves on what I thought was a PTO at my children's elementary school. I am insisting that we need bylaws and standing rules but other officers disagree because they say we are only a school support organization. From what I can see, a SSO organization is just a broad term for a PTA, PTO, or booster club. They say we are governed by the bylaws of our our Council but from what I can see there is nothing in them that pertain to us except the following. Under Article IV - Membership and Dues, it reads as follows: The Council will be composed of: The City Council Officers, each Local President and/or their delegates from each school with membership status in good standing. These are the voting members of the council.
We are a 501c3. At our individual we have member that have paid dues. Our present officers (2 Co-Presidents, a VP, Secretary, Treasure and Co-Treasurer) were basically appointed by staff members from the school and have just climbed up the ranks. There has never been an official election. It is extremely hard to get anyone to even volunteer let alone take on an officer position but still I feel we are not doing things properly. They all feel we just need to have a Policy & Procedures manual but I feel otherwise.
Any advice or direction would be appreciated. If it makes a difference, we are located in the state of TN.