In that situation, it’s all a matter of degree. 
“OK.”
To me it’s positive, but not exactly enthusiastic unless it’s delivered in a really emphatic way, in which case they’d probably add something other than sure, anyway, like sure I’d love to, or sure, that’d be great!
In these specific circumstances…
…I’d think it was fairly negative. “Sure” is what I’d expect for something not very special, like “shall we make spag bol for dinner tonight?” “Sure.” But for something where there’s effort and it’s somewhat special, like a show or a restaurant (unless you’re including MacDonald’s in restaurants, which they aren’t in the UK on a casual basis)? It sounds like she’s not particularly willing to go, really, and would only be going along to please me or to save the effort of thinking of something to do herself. So yes, I would assume that it might be a good idea to suggest something else instead.
(I’m not American) – picture which I generally get from this thread, is that in US: “sure” is a nominally-positive expression – weighted more than a little toward being, ironically, a negative one. I voted “something else” – perceiving that it can signify something worse than the worst in the list of meanings: i.e. not merely “this is an idiotic suggestion on your part”, but, flatly, “you are an idiot”.
I’m thinking of an exchange a few years ago, on another board: where I had put forward a fairly long but in my view, cogent and sense-makingly reasoned point of view on a particular topic; and received from a well-respected member of the board, the laconic response “Yeah sure dude”. Was tempted to respond in turn in a way which would have, on this board, got me at best sternly warned, at worst banned. In fact I just let it go; not wishing to get into a similar situation on that board. I continue, however, rather frequently to think uncharitable thoughts toward the bod concerned.
It means yes. Nothing less, or more than that.
As a person who responds in such a manner frequently, when I say it I mean something in the vicinity of “There’s a possibility that I might enjoy it,” “Meh,” and “I don’t like that idea, but I’m too polite to say it,” so I assume that’s what everyone else means, too.
Just FYI, I did specifically rule out sarcasm and similar tones in my OP
I didn’t read you as ruling out, the usual meaning being seen by oneself, as sarcastic and “upside-down”; maybe I misunderstood.
Perhaps I was unclear on it, but I meant to rule out intentionally “opposite” meanings, such as sarcasm. I am interested in the meaning of the word as expressed straightforwardly and sincerely, in other words with an ordinary tone of voice.
With this in mind, I believe a sincere “sure” is essentially “yes”, and thus I picked the first three options.
“Sure thing” is what I say via email or text.
Verbally, “sure” is not something I use to express an affirmative. Unless I’m being sarcastic.
I’ve heard people opine that a text reply “k” is somehow rude, and I don’t get it.
I’ll get a text asking if I wanna meet at a specific place at a specific time. I’m all into the idea, but I’m driving. At a red light I’ll reply, “k”. What’s wrong with that?
This is a holdover from when texts cost cents to send. Having to pay 10 cents to read a single letter was definitely rude. It’s still considered rude even with unlimited texting because you’ve deemed them not worthy of an entire word or even thought. In fact, I deem it pretty similar to my “sure” which is “I’m saying this so you shut up.” Also, nobody has any idea that you’re driving. If you let people know ahead of time, “hey, if you text me when I’m driving but I don’t want to be rude to you, I’ll text you a ‘k’ at a red light to let you know I got it”, that’ll smooth over a lot of people’s feelings because now you have an explanation for being rude.
But the thing is, even if I’m talking face-to-face with someone and they ask if I wanna go to Moe’s Bar for lunch, if I’m in total agreement, I’ll answer, OK. So if I’m texting, what should I do, copy/paste a random paragraph after my “k”?
Looking through my texts with my gf right now, there are dozens of situations in the last few months were she texts, “wanna go to X tonight?” and I reply “k”. No further communication is necessary.
I imagine your girlfriend understands your mannerisms pretty well, and that’s why she’s not offended. She knows what you mean. If someone gets offended, the reason why is they don’t know what you mean and aren’t that in tune with you, so they expect more words out of you. So yeah, in that case if you never get the chance to explain to them you text “k” when driving, the expectation is that you should use more words to clarify your meaning. “Ok, sounds great”. They get to hand off the convenience of an immediate but abrupt reply with a slower but more cordial one.
This reminds me of an almost identical issue my wife and I have regarding the word “fine”.
If I say “That’s fine” I mean it in a generally positive way to indicate that while there may be a number of alternatives that are equally good, this choice is no worse than any of them.
She takes it as a negative response indicating that I didn’t like the choice.
So if she tells me that she got a vanilla ice cream for me instead of chocolate, and I say “vanilla’s fine” I mean you could have gotten me vanilla or chocolate and I would be equally happy or may even have a slight preference for vanilla. What she hears is that I wanted chocolate.
Well, it would depend, wouldn’t it? If someone you were seeing were to text you to:
then that sounds like date-night activity, to me. So if you said k, it would sound half-hearted at best unless it was just one text among many in-person conversations.
Probably fine when you’re going out places a lot, which you seem to be. But most married couples who’ve been married several years don’t do that. There’s more effort involved in arranging the situa-
Aha! I have it! If you reply k or sure to something for which there is not much effort involved in the suggestion, or where all the effort went into former conversations between you and the other person, that is fine.
OTOH, if you say the same to something which took some effort, not a ton, but a bit, that’s a bit dismissive. If you expect someone to choose a show or restaurant for you then you can gift them, in return, with a full sentence 
Lived in England all my life, voted for the third option (didn’t realise it was multi-choice until it was too late, duh). But actually I’m happy I only voted for that one, which I would call “indifferent but leaning positive” - in reality, it can mean any of those things, depending on tone. As others have said. If in text form, I would consider it most likely positive, but with a possibility that they’re not really that into it.
“Fine” has always been an interesting word to me, in that it seems to have two distinctly different connotations, depending on where it occurs in a sentence!
When used before a noun, it usually connotes a positive: We talk about fine dining, or say that someone is a “fine painter.”
But when it’s used as a predicate adjective, it’s much more ambivalent. “How was the movie?” “Oh, it was fine.” In that case, “fine” means it was just okay. There was nothing egregiously wrong with it, but it wasn’t particularly outstanding either. It was just…fine.
Of course, as with anything, it can also be used passive-aggressively. “I’m fine” to actually mean “I’m angry as hell, but don’t want to talk about it right now.” It seems like your wife may be hearing your use of “fine” in that way, when what you really mean is the “okay” sense.
I voted “Something Else” for this reason as well. It’s language; and while it may follow guidelines, it’s still rather fluid and ambiguous. The word “sure” is very ambiguous ore me. It’s an acknowledgement and an acceptance of your request. If I’m tired it could mean “whatever you want, I don’t care.” Plus the other uses That Don Guy mentioned above.