Generally a tether is the electrical cable which goes into the ROV. It carries all the electrical power, signals, video and fibre optics required to operate the ROV.
If the cable is damaged, the tether requires a retermination (reterm)
I’m not sure what you mean by "floating" a TMS. Are you talking about carrying out a load test on a TMS? A TMS is basically an underwater version of the surface winch. The ROV docks into the bottom of the TMS, and the ROV and TMS are lowered to close to the seabed where the ROV then releases and the TMS pays out cable (tether) to the ROV.
The cable from the surface to the top of the TMS (the umbilical) has to lift the weight of the ROV and the TMS. In the water the ROV generally weighs nothing due to its buoyancy. When load testing the launch system and lifting point on the TMS, you have to calculate what the biggest load that will happen to the lift point on the TMS.
Usually, the biggest load is the weight of teh ROV and the weight of teh TMS when they are out of the water.
In deeper diving systems, the biggest weight will be the weight of the TMS (in water) PLUS the weight of the armoured cable which goes from the surface launch system to the top of the TMS (this can be a good few tons)
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