These are rather scary words.
The indication is that the person is unwilling to compromise. This can be beneficial, if the insistence is upon the Doctrine (shariat). In all probability above words are used in other matters, not the Doctrine. In one case these words were used as the title of an article in a publication of the type that is supposed to be published regularly but die out rather soon, after first few issues. The article was by a young girl. Hazrat Shaikh (DB) keeps telling us that today we keep asking for supplication for all worldly things but never say that supplicate for my son or daughter so that he or she can go to the Heavens with ease. The strange combination is that give modern education and supplement it by supplications. The means adopted should be appropriate for the goal. The goal is to reach Gennah. With ease. If our children insist upon the Doctrine then it say something about their upbringing. If they insist upon anything else then it says very different things about their upbringing. Today we have the freedom to insist upon whatever we wish to. There shall be a day when we shall have to render account of our actions. Including insistence.
"Was your insistence in consonance with My injunctions?" "Why or why not?"
Only if we could take heed while still there is time.