Confusion about when to use B*Ib = Ic vs the voltage diode equation for transistors to obtain Ic
All models are wrong, but some are useful
Neither of the expressions you give is correct, as they ignore the collector voltage, β is not a constant, and the temperature sensitivity is rarely accurately known enough. However, they are both useful.
If the base is being substantially current fed, so from a high impedance, then the β model is most useful. If we are substantially controlling the base voltage, so driving it with a low impedance, then the diode equation can be easier to use.
When we design transistor amplifiers, we need to be able to tolerate the large changes in temperature and β that would otherwise upset the bias conditions, so these approximations are quite good enough to tell us whether we have a workable design. Any precision work needs feedback, and that's a different set of equations.
When I design a transistor amplifier, I tend to choose the working collector current of each stage, then work backwards with the β formula to what base current (to what range of base currents) that would require, and then see how much voltage drop is caused in my proposed biassing network or feedback divider by that current. If it's too high, or too variable, then I can reduce the bias impedances, or choose a higher β transistor configuration, or otherwise iterate the design to tolerate the range.
I don't find a need to use the base voltage formula. It's not predictive enough to use for setting up bias conditions. When I need to know the response of a transistor to base voltage variations, I use a gm model (basically the differential of that), or better still, S-parameters.