I'm a lone parent too - absolutely no involvement from my kids dads.
What jumped out at me was your need to label what you are to each other. Does it really matter what you call each other - why not work out what you both want from each other. Do you actually know what you want ?There are young (and probably traumatised) kids here and you need to step carefully.
I do have a relationship now - but I have been clear about what it is I want/need/can offer someone and so has he. Not sure what you'd call it really though. I think this may be what you need to work out. What do you want, both now and for the future (without getting too ahead of yourself - do you want to marry or stay independent for example) ? What can you realistically offer to him in terms of time and attention ? What do you want back from him and what can he offer you ? I had to accept that my life will never look how I thought it would when I was younger and come to terms with that. Almost grieve for the idea of that perfect family and home. I suspect this may help you - you need to give up the idea that all relationships look the same and that you can make up your own "rules" between you.
I have been clear that I will not marry again, nor live with anyone while the children are at home if at all (and they are still young so we are talking many years here). My children are my priority and they have been through a lot. As such, I will cancel on seeing him if I need to. Anything I need to do for/with them, I will. I will not stand for any crap over doing that.
He also has children who have been through a lot and he has them way more than 50% of the time including all weekends. I think this helps him understand where I am coming from and also reduces his availability - which I understand and accept.
There is no link between our children so our lives are not enmeshed in that way and I am cautious about keeping things separate. All the children need lots of TLC and we try to ensure that they get it. Practically, it limits the time we can spend together but I'm lucky to have family help so I can pop out once the kids are in bed or we facetime a lot.
However, I am not into casual so if someone is sleeping with me I expect that they are not sleeping with anyone else. I cannot do a traditional fwb thing. I need to know he is there if I need him (and vice versa) and if we can help each other with anything, we will (from talking through problems to putting up a shelf).
It is hard, because sometimes it reminds me of what I cannot have. But it is also great having someone who understands what I am able to offer and feels enough for me to be able to accept that. He makes me feel calm, safe and happy.
I don't know if any of that helps - but good luck.