AITAH- leadership style

So I (38F) have been in a co-supervisor position with someone (42F) for about 6 months and things started out really great but have gone downhill since. There have been some mishaps between us, mostly due to our perspective and approach on leadership style. (I press the gas while she presses the breaks.) Some examples are me sharing a motivational video by Brene Brown, and she still hasn’t watched it. (I know I cant make her watch it, so I’ve dropped it.) Another time I asked her if we should set up a meeting to have EAP come talk to our team- she responded “no I have a lot on my plate.” Which is okay, I’m not going to judge someone’s load level. Our company merged with another and in our interview, we were asked if we were okay with learning the process of the other company. So when I started learning it, I had asked her if she wanted to join so we could learn together, she said “No, I’m good” Recently we had a mishap where I asked her if she was working on some reports, she said no, so I then started working on them and when I went to email who I needed to, I went back into the report and saw she was working on one. So I texted her saying, “When I asked if you were working on them, I was asking so I could know if I could start working on them.” I sent my email and added “I was working on this, and I now see that so-and-so is working on them.” Then she texted me and said I made it sound like she took over and essentially made her look bad. I did tell her I was bothered about how the situation happened, but that my intention was to not make her look bad but to acknowledge that I was aware she was also working on it.
Fast forward to the day after and I pulled her aside and said I could feel tension between us and wanted to clear up/reiterate that I did not intend what she thought I was doing with what I said in the email. I said, “I’ll work on how to improve the way I come across in emails.” She said, “ yeah, and to be brutally honest, I also feel like you think we are in a competition.”
I was shocked. I told that is not who I am and at the end of the day, I want everyone to win.
So, AITAH??