If I'm the one not interested in dating, I would still offer to pay. I want to be chivalrous and a gentleman. Instead of offering to meet again, I'll call her the next day nonchalantly and say:

Hey, was great meeting you. Take care!

I want to be polite.

If a date absolutely insists on splitting the bill the first time, this gives me a clear signal that she isn't interested in me and won't ask for a second date, but I would still call/email to contact me if she would ever want to go out again, not mentioning the word "date". Instead of calling the next day, I'll wait to the third day before contacting her. If mutual friends set me up on the date, I can push them to convince her that I'm quite a catch, the best guy in the world, and the next date will be even more awesome. If I'm told that I'm being "friend zoned", I will be "too busy with work" and avoid my friends for at least a month. I want to make sure I'm over her so it won't be awkward the next time we meet.

If I really would like to see her again, and she very much didn't appear interested, I would invite her to something really cool, but to "hang out" with a group of people familiar to her (a club we belong to, mutual friends, something public like a convention or some famous people I know) if she would like. I do have to keep the meet relaxed, hopefully making it more comfortable for her, in case I get friend zoned so it won't be awkward.

When we meet, I act excited to see her and move in for a "friendly" hug (short and not tight). From there, I instantly act nonchalant not to look too interested. I bring her a gift, something cheap (or free) and not gender specific. This can be something general in nature or something I think she would actually like. This could be something simple like a phone app, steam game (if she is into that), or something physical in nature that can fit in my pocket. What I say is:

Hey, saw this and thought it was cool and that you would like it. (pause if she doesn't take it after 5 seconds) Otherwise, ya know, I thought it was cool (put it back in my pocket).

I want this action to signal to her (stay close in personal space, but no intentional physical contact), even when we are meeting in a group, I don't want to be friend zoned. If she avoids me and hangs out with other people the rest of the night, and if we met there instead of picking her up, then after an hour or so, I'll "have to leave early" but thank her there instead of calling her the next day. I'm mature enough not to be persistent, but I will probably think of her as a bitch. Sometimes, for some strange reason, ignoring a lady for a couple weeks might make her more interested in me. I don't know why this happens, but if she contacts me out of the blue, it is pretty awesome.

Just a note, if we meet online (I never go on dating sites), I delete every sketchy comment I have ever written in history before asking her out. I have written soooo many sketchy comments that would make 4chan blush.