I am not making any statement about who is or is not sexist in this instance. I am solely talking about the role of intent with regards to actions that may or may not be sexist.

Also, note that that definition is not the one that most people who care about this topic use. That would be 'sexual discrimination.' Sexism is a systemic cultural force of sexual discrimination. That said, it's not particularly relevant to what I'm saying either, but I thought I'd mention it.