My RedBellied, Oscar, is a wild and crazy guy. His former owner "petted" him by toweling him to remove him from his cage and then stroking him with an oven mitt! For two and a half years, he has ignored me or bitten me whenever I offer him food or pets. Suddenly, he has begun to come over and seek me out for interactions, usually talking to him, giving him treats he will not eat or singing to him. Just recently he has allowed me to scritch his neck/head gently for thirty/forty seconds before he whips his head around to bite/nip. Usually it is a nip, but now and then he does "latch on" and not let go like your Gizmo. Oscar wants one on one interaction, but I don't think he knows how to react to it. He assiduously has tried to talk my Senegal hen and every cockatiel hen into pairing off with him while being dominant and physically aggressive to the cockatiel cocks. He ignores the love birds and budgies, but they give him a wide area when they pass him or sit near him. His behavior is very like my Meyers Parrot, Hobbs, who was a bite to the bone jealous bird when I first got him, but over a three year period was able to change his behavior with a behavior mod program I got from Sally Blanchard when she still made sense about bird behavior.
Gizmo's reaction to things on his table, was very much like Hobbs' reaction to anything which took my attention away from HIM when he was out of his cage. He would literally attack the object (magazine, phone, TV remote, etc) forcing me to drop it and then punish me by latching onto my fingers and grinding his beak until he got down to the bone sheath. I have about eight nice white scars from areas where he did this feat. I had to carry a tea towel on my shoulder just in case he starting his grinding bite because the only way I could stop his grind was to drape a towel over him and gently squeeze his body until he realized he couldn't take a deep breath and would then stop his grinding and let go of my skin. It took six episodes of this stand-off to teach Hobbs to not bite me this way! He was stubborn. But toweling him, removing him from my presence and putting him in the time-out cage until he calmed down worked to eventually eliminate his jealous behavior and produce a loving bird who enjoyed scritches and could be trusted to sit on my shoulder and nap three years after behavior modification started. Gizmo is definitely defending his table territory, but in addition, he has taught you to treat it like a game and it is his way of interacting with you. My first suggestion is to stop interacting with him on the table and on the top of his cage and remove him to a "strange" room or area where he has no "territory" to defend. Bring toys into the situation, things like a ball you can gently roll toward him and allow him to "fight" with until he burns off his energy and becomes more easy to interact with. Then introduce other interactive toys, like stacking rings for him to remove from a tower, etc. Eventually your goal is to teach him to stack the rings on the tower, but it is easier to initially use his energy to remove the rings, so you start there. Gizmo has two problems: he is high energy and he is territorial. You have to defuse his behavior before you can teach him to interact socially. And I am sure he wants that social interaction with you, but has no idea how to go about interacting appropriately so the interaction lasts longer than the thirty seconds it takes for him to whip his head around and nail you while you try and scritch his head!
It is a long and bite-loaded process, fraught with two steps forward and three steps back episodes, but it can and does work and he can become a loving pet if only you can find a way to get past his problems. Good luck with Gizmo and I very much hope you can break through his current behavior and turn him into a loving pet. I am making show progress with Oscar but I know I can win him over; after all, I have until either he or I kicks the bucket!