Comparison of coefficients in log(y_t) and log(y_t/y_0) LHS specifications in LP-IV

I would have a question related to econometrics. Likely not all the details are needed, but please bear with me.
My goal is to use local projection with an instrument to find out the response of an outcome variable $y_{t+h}$ (in horizon $h=1,2,..,k$) to a shock that happens at time t. (The shock is instrumented, other controls are included as well.)
In one specification I include the outcome variable in logs, as $ln(y_{t+h})$, and get that the coefficient corresponding to the shock is positive but decreasing as h horizon increases. I suppose the interpretation of the result here is that the outcome variable increases by $beta_h$ % in response to 1 unit shock compared to what it would have been without the shock in horizon h.
In another specification, I include the outcome variable as $ln(y_{t+h})-ln(y_{t-1})$, looking at the growth with respect to time t-1. Here I get that the coefficients are negative and are also decreasing in time. I guess the interpretation is that the growth of y relative to time t-1 value decreases by $|beta_h|$*100 percentage points compared to what it would have been without the shock after h time-units.
I’ve been wondering: how can these two results be reconciled? One says that the outcome variable is higher than it would have been without the shock, and if I understand things correctly, the other result says that it is lower. Is there any connection that could be made between the two results?