Why is engine anti-ice secured in cruise and climb when SAT is -40 or less? Yet in descent it is turned on with visible moisture regardless of SAT.
The idea is that at SAT colder than -40, it's "too cold for ice to form." (That is itself a long discussion, but that's the conclusion the rule is based on.)
In cruise below -40 SAT, you don't need the engine anti-ice turned on, even in "visible moisture" or what seems to be such. But, somewhere in the descent, you'll get above that -40 cutoff, and at that point you do need the anti-ice, and probably quite promptly. So rather than letting you start a descent with the switches off & hoping / assuming that you'll notice when the temperature gets above -40, the rule is to turn it on when you start the descent in visible moisture so that it is on when you enter the regime that you need it.
In a climb, you turn it on when the TAT drops to +10. If you're distracted & don't get it on until the TAT reaches +5, nothing bad happens -- ice still isn't forming. Even at a TAT of 0, ice isn't forming all that fast. In a descent, you don't have that same margin of time... at -35 SAT, ice can form really fast. So we turn on the engine anti-ice for any descent in visible moisture so as to protect against the bad day when your first cue that "oh, we should have turned that on already" is your engines coughing & failing!
-40C has been traditionally defined as the temperature at which you can't encounter water in liquid form (supercooled droplets). (Recent research has found supercooled water as low as -48C, but in any case, the industry uses -40)
So, no requirement to turn cowls on below -40 while temperatures are expected to stay at least that cold in the climb/cruise case, but on a descent, you are going to be above -40 at some point soon after you start the descent, so get the cowl leading edges warm right away, especially since they aren't going to heat up as fast with the engines at idle.
Cowl anti icing is more critical than wing anti-ice, because ice forming around the cowl inlet gets shed and is rough on fan blades.
At the warm end of the temperature range, a TAT of +10 is used because you can easily get enough temperature drop in the cowl inlet at high thrust settings from the pressure drop, so +10 is used to guarantee the cowl inlet will always be above freezing when cowl A/I is off.
Even on the wings, taking off with with temps a few degrees above freezing if ground deicing wasn't used, you can have run-back water on top of the wing freeze just from the pressure drop as the wings start making lift.