FT is T that is not bound to SHBG or other proteins.
Note that SHBG does not transport and release T, SHBG bound [SHBG-T] is not bioavailable. You will find many incorrect references to the contrary.
SHBG-T is tightly bound. Estrogens bind weakly to SHBG and SHBG can transport estrogens and release the estrogens throughout the body. Note that SHBG occurs in blood, but does not exist in tissues. So all T in tissues is considered FT. Secretions, such as saliva, can be used for hormone lab work and the results are considered FT, thus there is no TT in saliva testing. Many feel that saliva testing is quite useless.
Weakly bound T: T can be bound to other proteins such as albumin in the blood. Those are weakly bound and the albumin bound T can be release T to tissues and this is considered hormone transport. There are other proteins to which T can be weakly bound. Note that some males loose serum albumin levels as they age. This is thought to also reduce bio-T:TT to some extent. In reality, these guys have low T, and lower albumin is one of the catabolic effects of low T. TRT would be expected to improve albumin levels.
With age, we see E2 and SHBG increase. This tends to increase TT and that can be misleading.
Bio-T is FT plus weakly bound T. Both measures are technically useful. Bio-T and FT are roughly proportional. So you typically do not have any need to do both of these types of labs.
TT is FT + weakly bound + SHBG-T. Note that for a given production rate of T, or a given TRT dose of T, the more SHBG you have, the more TT you will have. In this regard, TT can be the wrong therapeutic target and can be a simplistic and inappropriate concern if one does not understand the larger picture.
From a practical point of view, lab results should be useful. So the results need to be something that others are sufficiently familiar with to interpret. Most here are familiar with FT and not with bio-T. So FT will be better from that point of view.
To put the whole picture together, you need estradiol numbers. With FT, TT and E2, there is very little need for SHBG lab numbers.
Note that some labs produce high ranges and results for FT compared to other labs. FT lab numbers always need the lab ranges for interpretation. In this regard, I expect that bio-T has the same problem and bio-T results will always needs the lab's ranges for interpretation.
In HPTA intact males, LH is released in pulses and diurnal patterns. This leads to changing FT levels. FT has a short half life. When you test FT in these males, the absolute number is not very important as the result is partly a factor of when the lab work was done. With TRT using transdermals, the swings in FT are more extreme and lab timing greatly determines the lab values. With frequently injected T, TT, FT and E2 levels can be very steady and FT lab results thus are more valuable in that case.