Whether you define marriage as between a man and a woman or marriage as between any two consenting adults or marriage as anything else, arguing about it is always going to come down to an argument about definition. I don't think there's any way to avoid that. There's not really a way to define marriage in a way that's completely agnostic as to its makeup and then argue from there as to what marriage is. So yes, the question ultimately comes down to who defines marriage and what they define it as.
You say "arguing". Simply asserting a definition is not arguing, because it does not contain an argument. An argument is a non-circular conclusion that follows from premises. You don't make an argument, you just say a thing. Also, while different people can define different things differently, all definitions are also not equal. Defining marriage to mean doorknob and doorknob to mean tanker truck and tanker truck to mean multi-platinum selling R&B artist Drake are definitions, but they are clearly nonsense definitions. Some definitions are more reasonable than others.
Allow me to offer my suggestion for squaring away the "definition" of marriage. When it comes time to actually discuss public policy, we generally come up with some sort of theory about costs and benefits to society, costs and benefits to individuals, and standards. I do not believe marriage has a "definition", I believe whatever it
is, gaining access to it should
have some set of
requirements that are rationally connected to its benefits and costs, and take a form to serve important public and private roles. This is why we have changed the requirements in the past, in recognition that the previous set of requirements were unduly permissive or limiting or achieved undesireable social outcomes.
This is also incidentally why we can argue that an ideal marriage is loving without requiring love, or that marriage as an institution promotes procreation and stable child-rearing without requiring procreation and child-rearing. We can explore the incentive structure for social benefits and costs without arguing that each purported benefit is a requirement to marriage. That's my position. It has the benefit of not requiring me to exclude gay folks or having to be inconsistent when I allow shit straight marriages.
But setting aside my arguments that marriage doesn't have a "definition", let's evaluate your definition: The "complementarity" thing is nonsense because it's incoherent on any level that isn't simply "We need to exclude all gay couples and allow all straight couples". The idea that a guy and a girl are the only marriage possible because (among other reasons, since you also bizarrely mentioned "emotional" complimentarity, which is just baffling) according to you one has a penis and one has a vagina.
Okay, why are a penis and vagina considered complimentary? Because you can put one in the other? That's true of a bedpost and an eyesocket too, but we don't require those for a marriage. No, what you are expressing is their use for procreation. But the requirement cannot be procreation, because then you exclude barren and postmenopausal couples, and you don't opt in the procreating and non-married. So then you construct some sort of scaffolding about "intent"--you still do the act even if it doesn't amount to anything.
You might say, as
the book "What is Marriage?: Man and Woman: A Defense" does--side note: that title has two colons in it, so you know it's a serious academic work--that a gun without bullets is still a gun, but a gun without a hammer isn't a gun. Of course that's a nonsense analogy that deliberately reduces a complex social issue into a dumb linguistic game. But I just mention it because the last guy on GAF who made a weird argument about some sort of Platonic / Aristotelian ideal form and complimentary body parts recommended the book, so... Anyway, for the time being, sure, barren couples are allowed as long as they have sex or "intend" to procreate or whatever.
But there are no shortage of sexless marriages. Most because couples grow apart and being middle aged is tough, but some because they never intended for marriage to be a sexual thing to begin with... like, say, a green card marriage.
Okay, fine, so you retreat to "well, even if you don't want to have complimentary sex and even if you don't have complimentary sex, at least you theoretically could have complimentary sex".
But then clearly you think John Bobbitt can't get married against since his wife cut his willy off. Talk about victim blaming. Fine, the new position is simple. If someone, at birth, had the right genetics to generally give them a penis, then they ought to be able to marry, at any point in their life, someone who, at birth, generally had the right genetics to give them a vagina. I guess those angry LBGT activists at some point are going to bring up genetic conditions or whatever. it feels like we're being backed into a corner.
It feels like you'd be better off just saying "God made marriage between straight people; gay people are lesser and don't deserve it". It's a logically consistent definition and it certainly doesn't play coy. Of course, it's also pretty cruel.