Your "multiplication method" for handling weights only works for linear regression. Virtually every software in the world should handle weights to linear regression. If you want to trick your software into fitting this "weight-multiplied" model for you, you must suppress an intercept term. That's because when you supply a vector of weights to the regression model, your software isn't smart enough to know this should be the first column of the regressor matrix, so you add a constant term anyway, and this results in a different estimate of the other coefficients.
What is nominally called an intercept term and a coefficient in a model is pretty inconsequential, computationally, the intercept is just a variable of "all ones". As a matter of convenience, most software automatically adds it. You have to go through separate steps to remove it (or equivalently to add something else in it's place).