well, a famous writer once wrote:
’Tis but thy name that is my enemy;
Thou art thyself though, not a Montague.
What’s Montague? it is nor hand, nor foot
Nor arm, nor face, nor any other part
Belonging to a man. O! be some other name:
What’s in a name? that which we call a rose
By any other name would smell as sweet;
So Romeo would, were he not Romeo call’d,
Retain that dear perfection which he owes
Without that title. Romeo, doff thy name;
And for that name, which is no part of thee,
Take all myself.
or in simpler terms; if it walks like a duck and quacks like a duck, chances are it's a duck
I'll leave it up to you to figure out if the simple lack of "HOA" in the title of the corporation makes then not an HOA.
Then, if it is determined they are in fact an HOA (btw: have you checked to ensure the rules they are attempting to enforce are in fact recorded in a state HOA Depository), then you could then demand they comply with the applicable laws regarding HOA.
If they are not an HOA, it's up to you to determine what they are. Then you can research the laws applicable to whatever type of organization and see that those are enforced.
If they have no authority to enforce the rules they are attempting to enforce, tell them to take a hike.
of course, never forget that an individual can seek to have the covenants enforced through the courts.