Well, you are right, he does think it's a game. Especially because he does something he's told not to and THEN he gets your lap and thus, your attention, even if he's upset.
In my opinion, the missing piece is some sort of 'containment' option which is not a lap or a 'naughty chair', where they can pop out and further engage you. Strollers are great for this, if you have a reasonably sized one. Bring it indoors, or if you have a pack-n-play, now is the time to set it up.
I went through the EXACT same thing with my son and I'll tell you what worked for us, step by step.
1. Give a clear choice: "You may keep your feet on the floor or you may sit in your stroller/go in the pack-n-play."
2. If feet stay on the floor, awesome. If he climbs on the coffee table: "I see you need to sit in the stroller/pack-n-play. Let's do that now." Then, put him where you need him to be-- the very FIRST time it happens.
3. Give him a few toys to play with-- this is not a punishment, you are teaching him to be safe, that's all-- and then go about your business and what you need to do. If he's crying, then let him out after a short time (about two minutes) , and if he's playing, let him play happily-- and get him out when he cues that he's ready.
4. When he is out of the stroller/pack-n-play, remind him "Feet stay on the floor now." And try it again. If he climbs again-- "I see you are not ready to keep your feet on the floor." and put him back into stroller/pack-n-play.
Here's the thing to remember-- it is essential to a child's development that they learn how to climb, so this is *why* they do this. That said, we should not allow it to happen in any place that is inappropriate. Be sure he gets lots of time at the park on safe equipment to learn how to climb. Consider getting a small slide for home. (some fold up). Let him know "climbing is for outside/at the park. Our feet stay on the floor."
All that said, keep giving him two choices *you* can live with. "You may do X (desired action-- "be gentle with your hands" "keep the playdough at the table" etc.-- tell him what you *do* want him to do, don't talk about what you don't want him to do) OR you may ("sit in the stroller" "put away the playdough" etc.)" He will tantrum. This is okay-- many kids do this. I myself do not try to talk a child through a tantrum; instead, I will try to give empathy as much as I can *before* the tantrum arises. "I know, you really want to climb. You wish you could climb right now." "I see you really want to X,,, it makes you sad that you can't do that." When he is tantruming, however, find a safe place for him to do it and don't try to fix it. He needs to get that energy and anger out-- he is likely not in a place where reasoning will work.
Keep the distraction in the mix, too. I have a five year old and it still works from time to time.:)
And for what it's worth, the reason I offer a 'containment' option is because we moms have stuff to do. Sometimes, I needed to make dinner or do housework....sometimes, we don't have the time to keep putting them back on the floor.