My wife and I watch a TV series called "Intervention", which portrays the lives of drug/alcohol addicts at their worst, at the point where it is destroying them and their families, ultimately ending in an intervention. It's tough to watch. Addiction and infidelity have a lot in common, in fact, I honestly believe that infidelity IS a form of addiction. They lie. They hide what they are doing. They gaslight. They get defensive as hell. They live double lives, and don't even resemble the wonderful people they were before the addiction. And more than anything, they seem to live in a fantasy world, one where the drug becomes more important to them than loved ones, or even their own dignity and quality of life. Sometimes, you'll see episodes where the addict meets up with another addict whose addiction is just as dire as theirs is, and they "fall in love" with that person. They then just reinforce the other's fantasy world story, and spend their days in lala land, usually resulting in one or both of them dying. But it's not really love, they aren't capable of that. You can't love someone else if you don't love yourself, and no one who loves themselves allows themselves to live a life of lies and deceit. They just want someone else to play the same game they are playing, to normalize their addicted thoughts and behaviors, and to make them feel "loved" regardless of their own self-destruction.
When it comes time for the intervention, the interventionist always separates the couple during the time at the recovery center. There is simply no way that one addict can heal while their enabler and partner-in-crime are there providing temptation and reason to stay in an addicted frame of mind. It's the same reason that teachers separate the two kids that can't stop talking in class. And what happens after the intervention is over (and successful)? Almost every time, the addicted couple that was so "in love" finds out that they actually had nothing in common other than the addiction, and break up because the other person is bad news anyway. The few, rare success stories occur when both people get help, and succeed, and work really hard at it together. It's rare as diamonds. But it happens.
I mention all this because I get the sense that your relationship is in a similar frame of mind. I don't know who your husband is and have never read any of his posts, so my only knowledge of him comes from this thread itself. That being said, in my unprofessional opinion, it sounds as if your husband has experienced some really significant family issues that may have left him with a fractured sense of what a healthy relationship looks and feels like. It was never modeled for him. He may have issues with feeling worthy of love, and may struggle to actually love himself. Again, this is just me guessing, but I make those assumptions based on my own experience as someone with severe FOO issues. I won't make any more comments on him other than to say this... I don't think he's capable of a healthy relationship, with you or anyone else, at this time. He needs to work on those issues in therapy. This isn't a crisis of faith, it's a crisis of self. Faith can guide him, but he has to take the footsteps himself. If he works through those FOO issues and learns what a healthy relationship and what a healthy self-view is like, then his world will open up, and then maybe he will be better able to accept both you and himself.
As a WS yourself, you already know you have similar work to do. You have to dig through those whys, and if you have FOO issues to deal with (most WS's do) then those need to be dealt with BEFORE you deal with the marriage and the infidelity. It's not a matter of importance, it's a matter of reality, as in, "you can't walk until you crawl and you can't run before you walk".
This was mentioned earlier in this thread, but I'll say it again. You have to let go of the outcomes for now. I know you love him and want to save your marriage, and we're all here to help support you in that. But you need to know that the work that needs to be done is going to take time, and you can't predict how long that will take, but it is rare for someone to turn around a lifetime of unhealthy relationship skills in just a few weeks or months. He's given you a deadline, and that's his prerogative, but it's not realistic. Imagine that you fell and shattered your kneecap. A broken knee takes time to heal. It doesn't matter how much money you throw at it, how many doctors join in, or how much motivation and positive thoughts you have. It will take months, at least, to heal. Now imagine someone telling you that your knee needs to heal in a week, or they'll leave. Well... how's that going to turn out?
That's my long-winded way of saying that you need to be willing to lose the marriage in order to have any hope of saving the marriage. The very best gift you can give yourself is to go get emotionally healthy. Figure out what it is in your head that allowed you to ruin your own dignity, self-respect and decency, and engage in an affair. Yes, it disrespected your spouse, but moreover, you disrespected yourself. Again, a person that loves themselves, doesn't allow themselves to do things that devalue themselves. So figure out what is eating at you and work through it in therapy. Learn to love yourself. And when you do, THEN you reach out to your husband (if he is also healthy and ready and willing) and you can build a truly loving, truly healthy relationship together. But he might not wait that long. And that's the part you have to accept.
To answer your original question, how are you supposed to prove you love someone when even they don't know what that looks like?
Let me ask you this. How do babies know their family loves them? Babies don't even have a concept of love, and they don't know what it looks like or how to ask for it. But clearly babies show us that they feel loved. Babies know they are loved because of how we treat them, how we interact with them, and most of all, what we sacrifice for them, or when we put their needs ahead of our own. We show love to babies by modeling it for them.
That's how you prove your love to your spouse. You let go of the outcomes (which are what YOU want) and instead focus on what he wants, and what's best for him, even if it is to your own detriment. I can't tell you how that will manifest itself however. For example, parents will work three jobs to send their kids to college. That's not because they love working three jobs. They sacrifice for their kids, because the kids matter more than their own ease of life. In the same way, when an opportunity comes to do something completely unselfish that will benefit him, and if the act comes from love rather than manipulation, then that simple action may be all he needs to know that he is truly loved by you. In short, if you love him, prove it. Not by words, but by deeds.