@Lth If you could see me reading your post, you’d see this 
. I’ve been a sitter on TrustedHousesitters for many years and would never see a need to ask for references. I’ve also been the first sitter for many owners and am happy to step them through any concerns they may have.
The pace and tone of this sitter, from what you’ve shared, would be off-putting for me. I’d see this as a red flag, and somewhat disrespectful towards you. If you do connect with her, I would ask what is her underlying concern that’s prompting her to make this request. I also would suggest a video chat, as this might allay any fears for both of you, or allow one of you to decide it’s not a good fit.
Personally, I’d swipe left and keep looking. I’d suggest you also embed your listing in your forum username/profile. Take every opportunity to promote your listing and perhaps get useful feedback on its content.