Hi @g3ml0u - i was in a very similar position to you 7 months ago. I'd been with my partner for around 8 months when I met his then three y.o. I met up with the child's mother first as both parents and I felt this was a natural step in doing what was right by the child. If you have the right circumstances to work with the child's mother, then I think that is such a huge advantage. She wants to know you're safe around her child, of course, and it's always easier when everyone acts like adults (although I appreciate how lucky I am and we can't control other peoples' behaviours!).
It sounds like you have totally the right attitude. You're already confident with kids (something I was/am not!) so that's a huge advantage, but also you are already thinking about the child in all of this - and that's the most important thing. My best advice would be to stay attuned to how the kid responds to you and go with it on their terms. Don't try to force yourself 'in', or to be their friend straight away. My partner's child has only a few days ago really started to show an interest in me (asking to sit by me, holding my hand and things like that), and we have hung out fairly regularly for 7 months now! It could've taken a year, or two years, if that's what was comfortable for them then that's all that mattered to me.
If you're serious about your relationship, then working towards a long-term goal of building trust and respect over time is so much more important in these early days than rushing in with big gestures early on and breaking hearts later down the line.
As for people getting petty about whether or not you think you're a step parent - the reality is, if you and your partner stay together then you will be eventually, and I think you're being responsible in taking this situation seriously. Because it is serious! And taking on a non-biological child is such a massive life commitment.
Best of luck, sounds like your partner is lucky to have an understanding person by their side!