To sum up the 3 general steps you have listed (Lets use a rock band as an example)
1. Make Song (Record Initial Tracks): This is typically the stage where the audio is actually recorded and the sampled/synthesized tracks are created. This is the stage that occurs in what most people call a "Recording Studio". In keeping with our example this is the stage where your goal as an engineer would be to capture good audio. The balance of tracks is less important here and you will be focusing primarily on capturing each individual track accurately. While some takes are done with full bands you will typically be recording one instrument at a time or possibly bass and drums together. Here you may use some EQ to get the sound you want but generally the tracks will be recorded as clean as possible.
This will occur in a place like this (Abby Road)
2. Mix Song: Now you have your individual tracks. Lets say 48 of them, drums (a bunch of ingle mic tracks and a few over head mics), bass, guitar, vox, piano and a few other percussion instruments. They have no effects on them and are individually really good tracks. You now will mix them so they become a coherent mix. Add in the effects you want maybe some more touch up with EQ and what ever else you want to put in. The end of this stage will be a mix near the final mix if not the final mix. Back in the day this very well would have become your master mix.
This will occur in a place like this (generally no live room attached)
3. Master Song: First off, this is not always done although although its common if you have the budget. In some cases this is where the vocal track will actually be added into the mix. In others a complete mix will hit the mastering room for further touch up. The mastering engineer may or may not have the multitrack recording but often will only have the master stereo mix from the previous stage. They will touch up with some EQ, maybe a bit of reverb or other effects to their liking and prepare a final release mix. Often the mastering engineer is taking what may be already a perfect mix and either touching it up or tailoring it for more of a consumer situation. Either way this stage does not always happen but has become more common. This often occurs in a mastering studio which is much like a listening room.
This will occur in a place like (note no console present)