May I come at this from a slightly different angle?
Modern quasi-religious language… and its slipperiness.
Your original post talks of "Bible-believing churches"and “doctrinally sound”. Subsequent language is of “inerrancy”. Our evangelical churches, very sadly, have an unfortunate habit of using such language to demonstrate tribal loyalty within our own echo chamber, and to proclaim our superiority over “the other”. (I wish this weren’t so, but alas, it seems so…)
Sure, there are classic doctrines from the early church, such as the Trinity and Hypostatic Union, for which such technical language is important. But it is important here precisely because it so so finely nuanced, and it is used with almost clinical precision.
But the more modern instances of pseudo-technical language (“Bible-believing church”, “doctrinally sound”, “inerrancy”, etc.) are very different. They are actually very poorly defined, imprecise and lack any nuance. Humpty-Dumpty in Alice in Wonderland springs to mind: “When I use a word…”
With these modern examples, pause. Take stock. Unpick. Examine.
“Bible-believing church”. Well, what do we really mean by this? I mean, really mean? Examine the counter case. It implies that there are “them over there” in what we critically judge to be in a “Non-Bible-believing church”. On what specific basis do I pride myself to be “Bible-believing” and judge them to be “non-Bible-believing”? Why, really, do I feel the need to draw this us-versus-them distinction? Pin it down. Pin myself down. Do I really not have a log in my own eye as I point out the speck in theirs?
“Doctrinally sound”. What “doctrine” is included in our judgement of “them”, the presumably “unsound”? Which doctrines do we think matter? (And why?) Which doctrines do we think don’t matter? (And why?) Who defines “sound”? (And how? And on what basis?) And how does any of this marry up with the classic creeds almost universally agreed as ‘core’ across almost all denominations? (Apostles’; Nicene (modulo ‘Filoque Clause’!); Athanasian.)
“Inerrancy”. Another recent invention, unrelated to classic church doctrines. What do we really mean by that? (For instance, what might it mean to apply inerrancy to “the kingdom of Heaven is like a mustard seed”?) The Chicago Statement, which claims to be authoritative on “inerrancy”, itself seems rather confused in places.
For myself, whenever I encounter such terminology being used, I always try to strip away its religious “command and control” aspect of “us insiders versus those outsiders”; to get it pinned down in non-religious language.
It can help if one tries hard to avoid such terminology; instead, re-cast it into something specifically and deliberately non-religious.
Hope that helps.