I agree with you.

I'm painfully shy and if I like someone, I still reach out. I'll find any excuse I can to talk to them. I was actually in your position once. Always initiating. Even though he was flirty and engaged in the conversation, he wasn't interested in me (he shot me down lol). He liked the attention I gave him (I was constantly stroking his ego). I assume he thought because I liked him, I'd be inclined to... IDK, make out, hook up. But he wasn't interested, he was just horny.

If I had to take a guess, I'd say she likes the attention you're giving her but isn't actually into you. However, I don't know you guys, no one here does. The only way you can know for sure is by asking her out or at least letting her know you fancy her. Her response will tell you all you need to know.