I don't think your mom is entirely off base here, other than the fact that kids usually find "stern" voiced funny, or giggle because they're nervous when you use them. But the eye contact (not in a menacing way, but in a 'being sure you have their attention' way) and repetition are your best tools at this age. That, and giving options for them to do instead of just what not to do.
If you don't want him on the couch, you're going to have to remove him every time you see him, and I'd recommend a bored, calm, but serious, "that's not safe, please stay on the floor." and repeat 7.2 billion times in the next 6 months.
Alternatively, if you're OK with him being on the couch but just not diving over it, I'd either try to find a way to move it permanently so it's backed up to a wall so he doesn't have the option, OR when he's on the couch say something like, "stay on the sittign part please" and then if he starts climbing to go over it, you say something like, "Keep your feet on the couch please!" while helping him down, and emphasize the point that so long as he's on the COUCH part he's fine to play, it's the climbing over it that's the problem. I had great success with saying "knees or bottom", while helping him do either, with our son to keep him from standing on furniture I wasn't OK with him standing on (i.e., dining room chairs) - living room furniture was OK with me, but not dining room chairs - and telling him what TO do was better for him than just what NOT to do. Again, you might have to do this a LOT dependingon his particular temperament and persistence, but that's totally normal for a toddler. Exhuasting for us, and I can't imagine having to chase around another same-aged toddler at that, so my hat is off to you!!