Rim compression after tire inflation I trued my wheel yesterday a bit, since many spokes lost some tension. Once I finished, all spokes were quite rigid.

Then I put the new tire on and as I was inflating it, I heard the spokes loosen up, with a typical cling, as heard while truing. The tension adjustment seems to have gone. 

The tire was indeed hard to put on, but after inflation it fits the rim properly, no part of the tire is sunk inside. The pressure is also correct, 0.4 MPa (the max pressure allowed being 0.45 MPa).

In theory, the inner tube expands equally strong in all directions, so also inwards; however I have not observed visible contraction of the rim so far, even in the cheap $15 models.

Maybe a wheel should be trued with the tire on?
The sounds you heard were the spokes unwinding as the pressure in the tire relieved some of the tension in the rim. As you tighten the spoke nipples the spokes tend to "wind up" due to the friction between the threads in the nipple and the threads of the spoke. If the friction is high enough (old corroded spokes/nipples or a lack of lubrication on the spoke threads) you're just winding up the spoke.

As you true a wheel, pay attention to the spokes – are they moving with the nipple, or are you actually able to tighten the nipple on the spoke. One place where you can easily see this is by watching the "back" of the nipple on the inside of the rim. Add some light oil if necessary to help lubricate the threads. Expect some wind up, it is pretty much unavoidable. To release the wound up spokes you can put the wheel down on a hard surface and press down gently on the rim. Work your way around the wheel pressing down on the rim – you should hear the spokes unwinding and perhaps feel them moving as well. Do it on both sides of the wheel.

That should go a long way towards resolving the problem. For what it is worth I never true new wheels with the tire on.