As stated in a comment, your understanding of protein translation in eukaryotes is not accurate. In order to provide a concrete reference that isn't wikipedia, I'll quote here from Biochemistry (2nd ed) by Voet and Voet (1995 / John Wiley & Sons) pg. 307:
Cytologists have long noted two classes of eukaryotic ribosomes, those free in the cytosol, and those bound to the ER so as to form the rough endoplasmic reticulum (RER ...) Both classes of ribosomes are nonetheless structurally identical; they differ only in the nature of the polypeptide they are synthesizing. Free ribosomes synthesize mostly soluble and mitochondrial proteins, wheras membrane-bound ribosomes manufacture transmembrane proteins and proteins destined for secretion, operation within the ER, or incorporation into lysosomes ... These latter proteins initially appear in the RER
Pretty succinct and thorough description.
The wikipedia text you quoted is a bit shallow (on its own) and somewhat incomplete, but essentially correct. The RER isn't solely associated with synthesis of proteins destined for secretion, but this is one of the classes of proteins synthesized in association with the RER. It would be more accurate to say that proteins destined for transmembrane insertion, secretion, or certain other kind of 'special handling' are synthesized in association with the RER rather than free in the cytosol.
The details of how, for example, these different classes are targeted to the RER are fascinating but somewhat beyond the scope of the question. Well explained in the referenced text, though, if you're interested.