If Brita can’t answer the questions about their differing models, then I wouldn’t have much confidence in them.
Total hardness (AKA GH) is the calcium & magnesium (responsible for scaling) in the water.
Temporary hardness is the buffer/bicarbonate/alkalinity/KH, in the water. This is the most relevant value.
TDS (total dissolved solids) is everything in the water.
TDS reading will usually be higher than the other two alone. You could add total & temporary hardness together and it exceed TDS, so they are terms for different parameters and there is no predictable relationship between the three.
Please check out one of the other water threads.