What is the difference between an independent contractor and a third-party consultant?

Legally, there is no difference between an independent contractor and a third-party consultant: an independent contractor is a third-party consultant. What their concern arises from is the fact that the law disfavors having "independent contractors" who fundamentally work like, are managed as, etc. employees: the law doesn't want employers to circumvent their obligatations by simply calling employees "contractors." Therefore, if someone works in essentially an employee-like capacity (employer manages or supervises them; employer sets hours; they work largely or entirely onsite or at another employer-determined location; etc.) they are, and must be compensated as, an employee.
If someone is brought on for a modest amount of time, it's easy to get away with treating them as a contractor: there is a pragmatic understanding that, especially in today's employment market, companies bring on extra help or expertise they need for defined projects or short periods of time without adding headcount. But if you keep someone on too long, you have to make them employee or else face potential penalties for "misclassifying" them.
To be a non-employee long-term, you need to not just set yourself up as a separate, independent business (an LLC, by the way, is better for this purpose than a DBA), but you also need to work on a more-independent basis. That is, you should set your own hours, as long as you get the work done by the specified deadlines; should be able to largely work remotely or offsite; and while the employer provides guidance as to what they want and evaluates the services or end product you provide, does not manage or supervise how you do the job. There has to be a substantial amount of independence in how you work, and, of course, not all jobs lend themselves to that.
Bear in mind that at the end of the day, it's up to the employer who to hire as either employee or contractor: while the above are general guidelines, it's up to this employer whether to give you the opportunity to keep working with them, and under what circumstances. Ask them whether there is some arrangement under which you can still work for them as a consultant; offer and be prepared to set up your own LLC and provide your own equipment/software (another thing that is looked at: independent contractors and consultants provide their own equipment; they don't use the employer's for the most part). There may or may not be some way they will still be willing to work with you, but you need to ask them what will satisfy them--and then see if that also works for you. Good luck.