You are correct that your `Model 2` makes the most sense if you wish to standardize your continuous predictor `C`. The confusion comes from what the intercept and the coefficient for the binary predictor `B` mean in `Model 0` versus `Model 2`.

I assume that Stata is using treatment coding of the predictors, and that by standardizing `C` you mean subtracting its mean and dividing by its standard deviation.

Then in `Model 0` the intercept (`_cons`) is the estimated outcome  when `C = 0` and `B = 0`. The coefficient for `B` in `Model 0` is the _difference_ from that outcome when `B = 1` and `C` is still at 0.

In `Model 2`, the intercept is the estimated outcome when `B = 0` and `C_S = 0`; equivalently, when `C` is at its original mean value. If `C` is associated with outcome and didn't have an original mean value of 0, that should be a good deal different from the intercept in `Model 0` even if there is no interaction with `B`, representing the outcome difference between `C` at 0 and `C` at its mean. So the change in intercept between models is expected.

Furthermore in `Model 2`, the coefficient for `B` represents the difference from the `Model 2` intercept outcome value when `B = 1` and `C_S` is still at 0 (or `C` is at its original mean value). If there is an interaction between `B` and `C`, then the magnitude of the association of B with outcome would differ depending on whether `C` is at 0 as in `Model 0` or at its mean value as in `Model 2`.  That's exactly what you're finding, and expected if there is an interaction.

You can find equations illustrating how simply centering a continuous predictor affects the intercept and the coefficients of other predictors interacting with it in [this answer](https://stats.stackexchange.com/a/417159/28500). If _all you did was center_ the continuous predictor, those equations show that the associated interaction coefficients don't change. For example, the interaction coefficient in `Model 0` between the continuous predictor `C` and the binary predictor `B` is the outcome difference following a _unit change_ in `C`, between `B = 0`  and `B = 1`. In a linear model, the effect of a unit change in `C` is the same regardless of whether you are starting from `C = 0` or from `C` at its mean value.

But in forming the `C_S` predictor for `Model 2` you didn't just center; you also scaled by the original standard deviation of `C`. So the coefficients for `C_S`  and for the `C_S:B` interaction represent the effects of unit changes _in the_ `C_S` _scale_, not in the original `C` scale. The coefficients associated with `C` in `Model 0` thus differ from those associated with `C_S` in `Model 2`, in a ratio determined by the standard deviation of `C` used in scaling to get `C_S`.