I used to think like that (and it does depend on the employer).
I was fearful that people would think i would have let them down or that they would use that as a reason to not promote me or not give me a payrise and over the last few years that they would use it as an excuse to make me redundant in the next round of cuts.
At somepoint though i realised that the people i work with arent like that and that when it comes to redundancies we've let some amazingly talented and hardworking people go just because thats where we are. I also realised i was over estimating my importance and irreplacability.
So me staying late, or not making an appointment wouldnt make the difference - its about what do i do when im there. Expectations are the problem more than anything (how often do we hear that????). And more seniority nearly always gives more freedom to make time for family
Do they believe it, is it just self aggrandizing or is it an excuse? doesnt matter, it wont change unless they feel it has to.
If you haven't seen it, id recomment watching 'the pursuit of happiness' (2006 film starring will smith) - its based on a true story.
Both mid 30s, 2 young kids BD 7sep14 XW moved on long ago, now living with OM1 D paperwork in progress