I’m not sure what they mean by “regular,” other than as a synonym for someone they routinely utilize.
A freelancer is a freelancer. They are not actually employed by the organization (magazine publisher, in this case). They are sometimes hired for a particular task (i.e. edit a particular issue or maybe be part of an ongoing subject), or they are permitted to choose from a list of upcoming articles and they are paid per article they submit.
Having said that, it sounds to me like YOU are one of those (if they accept your article or submission). They won’t hire you as a staff writer. You stay independent, and paid for however many works they accept from you. Might be on a per-word basis, or maybe per article.
That’s how I read it, anyway.