You can look on the Land Refistry to see if you are joint tenants or tenants in common, and you can change it if necessary for a small fee.
However, you should then still have a Will which sets out what you want to happen to the share of the house you each own, and to set up a Deed of Trust if necessary, and you should get a solicitor to do this. S/he would also check the Land Registry and can make the change for you, so I’d do that.
My Will, for example, leaves my share of our house to my sons, but with a life interest to my husband (so he can live there until he dies, or sell it, or rent it out) and names two Trustees (who are different to my executors). It explains what happens if he wants to sell, because legally he can only sell half, how the Deed of Trust applies if he buys another property etc. It also sets out what happens to the rest of my estate. His leaves his share split between family, with a life interest to me, etc.
So only our own shares count when assessing any care costs, as you say.
However, nobody is forced to sell or leave their home, whether they own their home jointly or as tenants in common, to pay their partner’s care costs. That only happens when the partner that hasn’t needed care dies, or if you are both in residential care and agree neither of you will never return home.
So a tenants in common arrangement is about protecting other people’s inheritance, not one person’s living arrangements if the other needs care. In some ways, it can be a bit more limiting, as you only have full control over your own share of the property. Plus, there is nothing to stop one of you selling or disposing of their share of the home regardless of whether the other one wants that, so you need to be sure that wouldn’t happen.
You should speak to a solicitor about whether it’s the best option for you both and about changing your home ownership model and Wills if it is.