Is it really CONSIDERED a batch effect when I extracted information from FASTQ file, reading first lines of it, and got the run_number and flowcell_ID from those lines? Or I am unintentionally reading too much by extracting such information for my RNA-sequence dataset which actually is not a batch effect?
It is not 'over-interpreting' - the information that you have extracted can indeed be used to identify potential batches. In RNA-seq, the sequencing run and flowcell are well known sources of technical variation / batch effects, and, for this reason, are sometimes explicitly included in the statistical model. The flowcell ID, in particular, can be important.
To check if these are actually driving a batch effect in your data, I would advise to generate a PCA bi-plot (or heatmap) of your normalised counts and colour the samples by flowcell / run. If you see a clear separation, then, yes, there is a batch effect. In that case, you can use this information as a covariate in your model, e.g., in DESeq2's design formula.