I'm sorry that your question wasn't answered in a timely manner. We don't have many avian vets on this site.
A bird that bobs its head up and down in a sort of pumping motion, beak open and then a purposeful delivery of partially digested food is regurgitating. This is something they would do to feed offspring or a mate. It’s done by some birds to objects (toys, mirrors, people) they are particularly fond of - especially if they’re in a breeding season, when a bird wants to please their owner and/or is of a nervous temperament.
Regurgitation that is unusual enough for you to make note of it like you have, might be a symptom of crop infection/impaction which threatens malnourishment due to loss of ingesta.
Vomiting is more of a head “flicking” event. The bird will often seem uneasy, pacing or uncomfortable and although the head bobbing might be similar to the regurgitation action, it’s usually more of a shaking and the end result is a very splattered, sticky substance that may or may not include food. When there’s blood showing in the vomitus it may indicate esophageal or proventricular ulcers.
An avian vet (please see here: www.aav.org) will take a look into your bird’s mouth as part of a thorough physical examination looking for ulceration or abnormal growths and is likely to examine a swab of the oral cavity for abnormal numbers of either bacteria, yeast, or parasites.
Vomiting is a more serious symptom and seeing a vet as soon as possible is recommended. Important considerations include ingluvitis (crop inflammation) due to bacterial, yeast, and parasitic (Trichomoniasis, e.g.) infections. Gram negative bacterial infections are commonly responsible for systemic infections in our pet birds. There are far too many possible diseases associated with vomiting to list here, but as in any case of illness, getting it evaluated, diagnosed and treated right away generally gives the best prognosis for recovery. I would be more comfortable knowing that Sunshine was examined by an avian vet - particularly because he's long past life-expectancy for a cockatiel and so degenerative disorders and age-related cancers become important considerations at this time.
If Sunshine won't drink and eat on its own please consider eyedroppering a few drops of the fluids and electrolyte replacer Pedialyte (or generics) every 20-30 minutes. Put the dropper gently inside his beak and let the drops fall into the bottom beak under the tongue rather than trying to get into the back of his throat. We don’t want to chance inhaling the fluid and developing aspiration pneumonia. Please heat up hisenvironment to 85F so he need not expend excess energy keeping his body temperature up.
Another feeding option is to offer all natural, organic baby food (squash, yams, sweet potatoes, mixed vegetables) which many birds take readily; also try some pabulum or baby rice cereal and a few licks of natural live-culture yogurt.
Nutritional imbalances are a common cause of illness in our pet birds. What has Sunshine's diet consisted of, please? Seeds should compose less than 20% of his diet. A diet of mainly seed and nuts has excessive fat, carbohydrates, and phosphorus; marginal protein; adequate vitamin E, and are deficient in amino acids, calcium, available phosphorus, sodium, manganese, zinc, iron, vitamins A, D3 (necessary for efficient absorption of calcium), K, and B12, riboflavin, pantothenic acid, choline, and available niacin. Ideally, a balanced pelleted diet such as can be found here: www.harrisonsbirdfoods.com or here: www.lafeber.com/pet-birds should be fed as well as hard boiled egg yolk, pancakes and cornbread, the tops of fresh greens, dairy products such as yogurt and cheese, fresh fruits such as apples, pears, melon, kiwi, and berries, vegetables such as broccoli, carrots, beets, asparagus, cabbage, sweet potato, and squash, and even tiny pieces of meat.
Please respond with the additional information and further questions or concerns if you wish.