Yes, this is indeed what is happening. The null hypothesis is different when you set the fold parameter.
For a DESeq2 analysis, the fold value is sent using the lfcThreshold parameter to DESeq2::results. DESeq2::lfcShrink is used to compute fold changes independently of p-value testing (but will respect the fold setting).
For an edgeR analysis, if fold is not zero, edgeR::glmTreat is called with the lfc parameter set to fold; when fold is zero, edgeR::glmQLFTest is used for testing.