I think you're confused by the nomenclature, because "elastic" and "inelastic" aren't your only two options --- there's a middle ground where most real collisions live, which is called "partially elastic" or "partially inelastic" depending on your mood.
An elastic collision is defined as a collision where the total kinetic energy of the interacting objects is the same before and after the collision. When you have two objects that collide, you can measure or compute the total kinetic energy before and after the collision; if the energy is conserved, you say "that was an elastic collision" just because "in this collision the total kinetic energy was a constant" has too many syllables. A completely elastic collision is a special case.
A completely inelastic collision is the other extreme of the special case, where the two objects end the interaction with zero momentum relative to each other. This represents the maximum energy that can be lost, because there's a reference frame where the final state of the system is at rest. Completely inelastic collisions are relatively easy to produce by making your projectiles stick together, as you point out in your question.
Real collisions generally live in the middle. You're right that a real collision makes sounds and generates heat, and that this energy comes from the initial kinetic energy of the system. However if the energy that's lost is a small fraction of the total initial energy, then we can say that the collision was approximately elastic, which makes the analysis easier.
In quantum-mechanical systems that don't have any internal degrees of freedom, it's possible to have a completely elastic collision. But macroscopic collisions always lose some, perhaps negligible, amount of energy.
(Sometimes you even hear about "super-elastic" collisions. These are collisions where some internal energy source is converted to kinetic energy. For instance, a gun is fired by striking the ammunition so that the gunpowder ignites and the bullet is propelled from the barrel; the collision between the moving part of the trigger mechanism and the bullet at rest is super-elastic, because there's more kinetic energy after the gun is fired than before.)