Basically it boils down to this:
You either use a descriptive or a prescriptive (normative) model, i.e. you either think that the market is always right or you think that you alone know how to determine the "true" price of an option.
The original idea of BS was to build a prescriptive model but most modern models try to take the market prices as given and calibrate their models accordingly, thereby gaining one common variable to make all prices comparable: volatility. One prerequisite is to seek consistency for all prices which is normally done via hedging arguments (which often leads to risk-neutrality assumptions). Yet in practice even in the case of minor inconsistencies they often cannot be used to make money due to friction.
At the end it all boils down to your business model (this is what I meant with my question about the true value of a stock). When you are on the buy side you try to find mispriced options with a prescriptive model (so you are saying that you know which is the true price - and all other market participants are wrong). When you are on the sell side you use a calibrated descriptive model which mainly seeks consistency so that you can live on the spread.
(This is why I gave this link in the comments: How do we use option price models (like Black-Scholes Model) to make money in practice?)
So in a way truth is relative: It either means overall consistent with the market or it means smarter than the market.