The linked article talks about irrigated agriculture so I'll focus on that, but things are generally the same for rain-fed agriculture, except you have less control on the water application and there is no loss in getting the rain to where it hits the ground (yes some rain evaporates as it falls but we measure rain at the ground).
For irrigation water, though some is lost between the water intake and the field. We don't usually consider evaporation along rivers but when you take water out of a river some evaporates as it flows through canals to where it is used. Some water is also lost through leakage since many canals are just excavated in the soil. Relatively few irrigation water systems are piped but that is one way to make water use more efficient. Lining canals with cement or plastic can help prevent leakage (a number of channels are lined with plastic near where I live but animals like koalas can get trapped as they try to cross).
The next place water is lost is in evaporation from the irrigation spray or ground surface. That is usually considered in calculating water use for growing crops. Hot dry weather will increase the loss and for some soils a dry surface can inhibit infiltration of the water. Some water usually ends up back in the streams or wetlands from surface runoff or from returns to the canal systems.
Another loss is water that flows through the root zone before plants can use it. That flow can be important in keeping salts from building up to the point where the crop yield declines or plants can't survive. On the other hand it can leach metals from the soil down into the groundwater.
As you probably know, plants take up water from their roots and use it in photosynthesis to grow. They also respire at night when there is no photosynthesis. That water is the major part of the water use by the plants - very little is left in the biomass. There are some ways to breed plants or engineer your crop (like pruning trees) to make the plant water use more efficient in terms of how much product yield you get, but these are somewhat limited. And, of course, water is used by weeds.
Bottom line, most water in agriculture ends up as evapotranspiration (evaporation from the surface plus transpiration from plants). Some is lost to the groundwater or returned to surface water from the irrigation systems.
There are further inefficiencies in meat and dairy production. Some water is lost in metabolism, urine, feces, etc. And animals have to eat a lot of food to put on weight.