The OP owns the house jointly, in equal shares. She's explicitly said so. 50% of the equity is hers.
She's also said that he is a far higher earner, and most of her own earnings have gone on paying all the children's expenses. She earns less because for a long time she worked part-time, caring for the couple's children.
They are a family. Two teenage kids, a family home that is half the OP's, a long life together. She has paid most of the kids's costs, he has met overheads for the house and bills. Not unusual in many, many marriages. She earns so much less because she took a career break. Why is that contribution being completely ignored?
The answers here are based on her being 'lucky' to be offered less than half the equity she actually owns, after a 20 year relationship and the lion's share of all childcare and child cost responsibilities.
He can afford to buy her out and live there. She can't afford to do anything but rent, to keep a roof over her children's heads for the remaining 4 years in school. She's simply asking for him to pay half the cost of the mortgage on an appreciating asset, while she covers everything else, until the kids have left, and then they can sell and she can get her half, and so can he.
He's got no costs in this purchase. No searches, no surveys, no removals, and will be able to remortgage rather than purchase mortgage, too. He gains from this financially, yet he's placing all those saved costs on her side of the balance sheet, by offering her less than her legal share. He is gaining the equity he isn't paying her, on top of retaining his own 50%, when he's fully aware all she will be able to do is rent - step off the housing ladder altogether. And women here would bite his arm off? Really?
It's half her house. They own it in joint shares. She has contributed to the family home because she was the primary domestic carer. No marriage means that isn't recognised in law, so it's lucky they own the house jointly. But why people are saying she should be grateful to be offered less than she owns in equity, she she will have to leave her family home and rent, after taking a significant lifetime hit on her own earnings to raise shared children, is beyond me.
OP, you need proper legal advice. Emily Watson at Raydens is amazing. She's also super kind. I would book a one-off advice session (it won't be cheap, because 'free' initial chats are sales talks, pure and simple) for her to go through where you stand. Then you can decide based on legal facts, and not a lot of knee-jerk, and at times frankly misogynist, opinion on MN.