TJI is a brand name for Weyerhaueser's i-joist product. They used to be owned by Truss Joist Macmillan, hence the "TJ", so they're TJ's I-joists or TJI's. Weyerhaueser isn't stupid when it comes to marketing and product awareness, so they kept the TJI moniker.
TGI isn't a brand to my knowledge. A lot of people mistakenly replace the J with a G...Very common. TJI was an innovator in the field, so they're used as a descriptor for I-joists in general just like people calling gypsum wallboard "sheetrock". Sheetrock is a brand name.
Other brands of I-joists include BCI's, which are Boise Cascade's version, and LPI's, which are Louisiana Pacific's (now called Solid Start).