It's either trust or arrogance that lets her feel comfortable enough to leave her horse in the care of a veritable stranger; perhaps she's naive, or perhaps she's just confident in her ability to decapitate anyone who'd do something as idiotic as turn that moment into a crime. (Spoilers: it's the latter.)

Integra is privately quite pleased to get out for a bit, even if it's just to eat food that isn't Hellsing kitchen sandwiches and sit on the wooden porch of a wild-west looking smithy and watch a man who looks like a bloody Hun shoe her horse. She catches herself thinking that the sun is irritating, for a moment, and promptly squashes the thought with prejudice. Once she's done eating, she pulls her CiD and a notebook out of a saddlebag, and is content to work quietly while Seoraj works probably rather loudly.