Clustering after PCA: Use the standardized data, or take into account the variation explained at each PC?
I had a more careful check, and I will try to answer my question based on the additional insights. I would appreciate further answers/comments regarding the accuracy of the provided answer.

The actual results of the PCs are not standardized, and their variance is proportional to the square root of the total variance (variation) explained by each PC. These values are the ones that should be used for the subsequent clustering. In this way, the weighting of the PCs is indeed proportional to the (square root of the) explained variance.

If the standardized PCs (mean=0, std=1) are used, then the values should be adjusted with the square root of the % of variance explained by each PC. In this way, the adjusted PCs will have the same weighting as the actual PCs, and the only difference will be on the range of the values.

I run the analysis for clustering based on 1) Actual PCs, 2) Standardized PCs, 3) Standardized PCs adjusted with the % of the variance, and 4) Standardized PCs adjusted with the square root of the % of the variance. 1) and 4) were practically equal (I assume some small differences are due to the fact that the range of the values affects the instances very close to 0) and the best ones, followed by 3). Option 2) was by far giving the worst performance. The performance was tested by the Sum of Squared Distances between each field and the allocated centroid on the original dataset (daily gridded data).