Order of batch effects removal, data imputation and library size normalization in scRNA-seq data
MAGIC assumes input data has been both library-size normalized, and either log or sqrt transformed prior to imputation (see also: MAGIC tutorial). Additionally, any graph-based methods (MAGIC, PHATE, t-SNE, UMAP, spectral clustering, Louvain, etc etc) will give flawed results if your data contains a batch effect, since the neighbourhood graph would reflect that structure of your batch effect, and worse, imputation would further reinforce this batch effect.

Thus I would recommend the following pipeline:


Library-size normalization
Square root (or log) transform
Batch effect removal
Imputation


Regarding your update, the reason you don't see the batch effect in the raw counts data is simply that the batch effect is not visible in the most highly expressed genes. Prior to transformation, the principal source of variation in your data is simply the expression of the most highly expressed genes - this is essentially masking the source of the batch effect, not removing it. I recommend never working with raw molecule counts in scRNAseq, as the raw counts data hides much of the heterogenity in your dataset, which is precisely what you are looking in doing single-cell RNA-seq.