so an electromagnet could be also called a solenoid 
You can call it what you like

So far we have inductor, relay, solenoid and now electromagnet
A generalisation for each of them could be something like this
They're all inductors, hell even a piece of wire has inductance but generally speaking when referring to an inductor, you are referring to a device that provides an inductance value and that inductance is affecting the electrical signal, by design or by frustration

A relay has a fixed core within a coil that when energised will attract an armature that does something, such as switch contacts and if the contacts are big enough you can call it a contactor.
A solenoid will have a moving core. When the coil is energised the magnetism pulls a plunger into the coil, the plunger is usually attached to some mechanical actuator. In the case of a starter motor, it moves the starter motor's cog into the flywheel gear and makes a large contact to provide current for the starter motor. Starter motors are normally series motors who's terminating speed is when the armature throws the windings out in a big sparkly mess... So its good to have it not start until the cog is fully engaged
An electromagnet will generally have a fixed core and nothing else. When the coil is energised the core becomes magnetised and that's it. These tend to be DC only. Metal scrap yards tend to use them to pick up metallic waste. In steel making plants several of them are placed in a row to pick up large metal slabs. I'm sure the magnetic field generated by electromagnets have many more uses but in some instances they may be called something different again, such as field or winding...