Is there an intuitive explanation for why the interpretation of the coefficients in a first-differenced regression specification is the same as the for the coefficients in the specification in levels? In a regression of say house prices on income, a first-differenced specification essentially regresses price growth on income growth. Why does the interpretation of b1 remain unchanged and does not switch to the effect of income growth on price growth instead of income on price? How could I actually investigate the former? Does the interpretation change if I add an additional variable in levels, and if so, why intuitively?

log(price) =b0 + b1*ln(income) + u
b1 – a 1% increase in income is associated with a b1% increase in (house) price

Δlog(price) =b1*Δln(income) + u
b1 – same interpretation as above

If I add an additional variable in levels, say a share or interest rate variable, does the interpretation of b1 go back to growth rates?
Δlog(price) =b1*Δln(income) + b2*irate + u

I have been scratching my head for quite some time on this but have not been able to find an intuitive explanation in any of my econometrics books.