I was really confused which piece of a chess game a "bishop" is supposed to be and had to do some googling, and it turns out: The bishop is only called "Bishop" in English and - apparently - Icelandic. I know it as "Läufer", which is the German word for "Runner" and it seems like most other languages translate it as some form of "Runner" or "Messenger". In French, it is a "Fou" (Jester, Fool) and in its Persian, Arabic and Indian origins, it was supposed to be an "Elephant". The confusion comes from how the piece looks, it's two sharp points on top being interpreted as tusks, a jester's cap or - in English - a mitra, the headgear of a bishop. (I wasn't able to figure out where to idea of a "Runner" comes from, unfortunately).

So now that we know that the bishop is a phenomenon only known in England, an answer to your question seems to present itself: For the majority of the Middle Ages, the relationship between the Catholic Church and the kings of England was strained at best, and if we look at a short description of chess from the 13th century (around the time when the "Bishop" is first mentioned in chess) we can read the following: "They [Bishops] move and take obliquely because nearly every bishop misuses his office through cupidity". So it seems that calling these chess pieces "Bishops" was - at least for some - a hidden insult to the unpopular high clergy in England. Of course, the bishops still stand next to king and queen on the board, symbolising a place of honour and power. So my guess is that both interpretations were common and used depending on the situation: When looking at a chess board, bishops could see a piece representing themselves next to king and queen, a statement to their influence and power. Someone else might look at the board and see the clergy meddling in warfare and politics, silently cursing them.

Link to the translated version of the chess description I mentioned:
https://archive.org/details/AHistoryOfChess/page/n553