Setting the conditions for the following operation is what "shaping" is, in this context. I mentioned that in a post last week, as regards the methodical series of strikes and disabling bridges, etc, taking out the Russian airfield's assets in Crimea.
"Softening up" is (IMO, and based) too vague of a term and more "World War II ish" - but maybe that's me being a bit pedantic. These are 'terms of art/jargon' used by the military.
A way to shape a battlefield, for example, is to drop a couple of bridges behind your opponent's forward line of troops so that the reserves, or reinforcements, can't respond as quickly when your own troops cross the line of departure and engage with them. It can done using air, artillery, or even a Spec Ops/SPENAZ kind of operation.
For a Hollywood example ~ yes, add a grain of salt ~ the Guns of Navarone special operation was a shaping of the battlefield, in terms of removing the guns from the battle so that the warships could do that thing they needed to do. For a more real life 'shaping of the battlefield" example, one of numerous attempts to shape the battlefield around Vicksburg, besides the various canal building attempts Grant undertook, would be around Steele Bayou.
There is more to IPB than that. Quite a bit more, to include terrain, forces (yes, weather for sure) avenues of approach, yadda yadda.