I'd take issue with that definition of due process as being a "general" description of it. All it really means is that, whatever the formal process is for legal disputes in society, you have the ability to avail yourself of it. And what those procedures are exactly will vary from society to society. Here we've set up a formal adversarial process, and included things like the right to examine witnesses and evidence presented against you, the right to present evidence and witnesses of your own, and so forth as being a part of the process. But this doesn't apply to other societies with other formal or informal procedures. In a society where guilt or innocence is determined by having the accused walk on hot coals, "due process" just means that you have the right to walk on hot coals to prove your innocence.
In a general sense, "due process" means that, whatever the formal procedure is, it is followed in your case, and you have the ability to avail yourself of it, instead of society just dealing with every case on an ad hoc basis. And if there is no set procedure, there is no real due process in any meaningful sense...
I tend to agree with Justice Brennan's assessment that the Fourteenth Amendment is best viewed as "a broadly worded injunction capable of being interpreted by future generations in accordance with the vision and needs of those generations." Oregon v. Mitchell (1970) 400 U.S. 112. I'm not at all afraid that future generations might conclude that "due process of law" requires for their time and circumstances somewhat different practices and procedures than ours. They should. The unifying priniciple is a respect for the individual's right to fundamentally fair procedures and a protection against arbitrary conduct on the part of the government.