The first point to discuss is that the normative halakhic practice throughout most of Jewish history is that while there are varying levels of forbidden, there is only one level of permissible. In other words, once something is permitted, it's completely permitted, with no reservations. So a piece of food that was produced by a process bedi'eved is just as permitted as food prepared l'chatchilah. I'm not trying to say that you are allowed to do everything in a way that's only kasher bedi'eved, but attempting to highlight one should separate the process from the result. Once the result is permitted, it's equally as permitted despite the process that brought it into being, but when you do the process, you should attempt to do so in the most correct manner. This is why the page you quoted says " it is 100% permissible to eat" because once it's permissible, it's 100% permissible. Whether your Rav would suggest you go through the process to do this is another matter.
As for whether this mixture is prohibited, the standard halakhic baselines are as follows:
The Shulchan Aruch (YD 95:1-2) rules that if a pareve food was cooked in a totally clean dairy vessel, the food may be eaten with meat, and that if a pareve food was cooked in a totally clean meat vessel, the food may be eaten with dairy.
The Rema argues against this and rules that pareve food cooked in a dairy ben yomo vessel (i.e. a vessel used with hot dairy in the past 24 hours) may not be eaten with meat, and vice versa, but that b’dieved, if such foods were already mixed together, they may be eaten.
Thus, according to the Shulchan Aruch (and therefore Sephardic practice), pareve soup cooked in a totally clean dairy pot could be eaten with meat, whereas the Rema would prohibit Ashkenazim from doing this as a practice, but allows that if you find yourself in a situation where the foods were already mixed, they are permissible to be eaten.
Many Jews take extra chumrot or follow other Rabbinic opinions on this matter, which is where you get lines such as: "If it's pareves cooked on meat equipment, you can't eat it with milk.". Rather than rely on such a chumrah, it would benefit you to read the relevant texts for yourself, come to a decision about where you feel the truth is in this matter, and then consult your Rav for a final decision.