What type of oolong is Twining’s oolong? Or what would this be closest too

I believe that 'English Breakfast Tea' is a very broad definition and doesn't refer to an exact blend; each manufacturer has their own version of it. I would say that English Breakfast Tea often includes Assam, but as your Wiki source suggests it also contains other teas. Therefore Assam is a separate tea.

White, green, oolong, and black tea are all products of Camellia sinensis leaves and buds, the only difference is how they are processed. You can turn the plant matter into any of the varieties. Different sources will give different accounts of the processing steps, but in rough order of least-to-most processed:

White tea is minimally processed and not oxidized, meaning it retains the natural antioxidants, but does not develop as much flavor, color, or caffeine. Sweet or flowery flavors are characteristic. (Chinese Silver needle white tea)

Green tea has a minimal amount of oxidation, halted by additional pan-frying (Chinese teas) or steaming (Japanese) steps. The additional processing brings out more flavor, and allows for caffeine to develop, at the cost of a reduction in antioxidants. Characteristic flavors are grassy, vegetal, and earthy, with sweet notes.

(Japanese Sencha)

Oolong tea gets partial fermentation, and often an additional shaking/bruising step that release additional flavors. Oolong can range in color from dark green to black. The combination of bruising and partial fermentation give many oolongs distinct flowery & earthy flavors. Darker oolongs may have buttery or smoky tastes to them. It has somewhat more caffeine and less antioxidants than green tea.

Tieguanyin or Iron Goddess: (light oolong)

Black Oolong: (Wu Yi)

Black tea is fully fermented, which blackens the leaves and causes the formation of caffeine and tannins. It generally possesses the most robust flavor and highest level of caffeine, but the least antioxidants.

(Chinese keemun)

Pu-erh is tea that is pressed into cakes and post-fermented, meaning that after processing and drying it undergoes an additional ripening/fermentation. Microorganisms work on the tea, changing it chemically in a similar fashion to aging cheeses. Traditionally this is accomplished over years in controlled climate conditions (10-15 years being ideal), but in recent decades a several-month fast ripening process is sometimes used.

Pu-erh develops a richer, more complex flavor, and has the widest range of flavors of any of the varieties. Before post-fermentation, it may resemble a white, green, or oolong tea, but the ripening process can impart darker black-tea flavors, giving a unique character.

Pu-er Cake

Rooibos, sometimes called red tea, is made from another plant entirely, Aspalathus linearis. It is a traditional South African infusion, prepared like a black tea but without caffeine. There are two varieties, green and red. Green is un-oxidized, where red is oxidized. Rooibos is characterized by a sweet, woody/malty flavor. Health-wise, it is a source of anti-oxidants, may aid in digestion, and some say it may aid with nervous tension and allergies.

Yerba mate is from the South American plant Ilex paraguariensis, whose leaves are steeped like tea. It is generally toasted in processing, and produces a brew with caffeine similar to coffee. Despite its caffeine, mate is actually relaxing to smooth muscles, so you don't get jitters, and it also helps control appetite. The flavor is vegetal and herbal, and can be bitter if brewed hot. Although it contains antioxidants, this is balanced by traces of carcinogens from the toasting, and the jury is still out on whether mate overall increases or decreases cancer rates.

Tisanes or herbal infusions these are various blends of dried herbs, fruits, spices, and flowers which are steeped like teas, but do not contain tea leaves. Generally, the ingredients are dried, although fresh ingredients can be used too. To confuse matters further, black and green teas may also have flavorings added to them, giving a similar result.