I think this is an issue where everyone has a different opinion based on their experiences.
I am a step-mum - my DSC have been in my life since they were 3 and 5 years old, and they are a massive part of my life. My DP and I share custody 50/50 with their mum.
I think the moment I became a step-mum was when DSS broke his arm falling off his bike, when his mum, his dad and me were sat in the hospital. We all had a moment of realisation at this point that any squabbling over titles / roles was irrelevant as he meant the world to all of us.
His mum does not refer to me as 'step-mum' but as "dad's girlfriend" (to school, friends' parents etc) - but both DSC's refer to me as step mum.
At least I am now 'dad's girlfriend' and not 'dad's new girlfriend'..which I was for a long time!
I refer to myself as a step-mum because after all these years it is not enough - not a big enough word to describe it - as my partner has 2 kids who I live with and care for 3/4 nights a week.
Sometimes you need a word like co-parent or step-parent to help other people understand that it's more than dating someone with kids (i.e. totally separate from your life).
However I do completely understand why the mother would not feel the need to broadcast the fact that her DC have a second 'mother' figure in their lives.
When DSS moves in with me and DP... well, that's a whole different ball game... "full time step mum" seems like the best option there - full time to one, part time to another... minefield!!
I think people (expecially exes) need to bear in mind that another woman loving and caring for and raising your DC's is not a bad thing, and who wouldn't rather (if they had to have one) that it's one who does love them rather than barely tolerates them?
Just my opinion...x