The deed is the instrument by which the former owner(s) grants or passes his/her/their interest in the property to the new owner(s), and if you are shown as one of the grantees on the deed, then you actually have an ownership interest in the property.
A "deed of trust", however, serves a different function, and is actually the legal instrument used in "deed of trust states" instead of a mortgage. It accomplishes exactly the same purpose as a mortgage, and in fact is often referred to as a mortgage in those same states because it is a term that the general public, and even much of the real estate community at large, is more familiar with. But no matter whether you're in a "deed of trust state" or a "mortgage state, these instruments are accompanied by a "note", which is the instrument that actually creates the legal obligation to repay the lender.
It's not uncommon at all, for a variety of reasons, for both members of a married couple or domestic partnership having the same legal rights as a married couple to be named in the mortgage or deed of trust, and in the deed, but not on the note. The most common case is that the one who is not named on the note has a credit score too low to meet the lender's credit requirements, but both want to be involved in ownership. In most states, if you are married or a member of such a domestic partnership, and your spouse/partner wants to mortgage a property, then it's a legal requirement that you sign the mortgage/deed of trust to provide you with "constructive notice". In other words, once you sign the mortgage/deed of trust, you have been legally notified that your partner is taking a loan against a property that you may have an interest in, and this whole concept is designed to protect the non-borrowing spouse/partner.
And regardless of all the above, you cannot be held responsible for the repayment of the debt, created by the note, unless you signed the note. But if the payments aren't made on time to satisfy the requirements of the note, then the lender can foreclose, and actually take the house from you, regardless of your ownership interest.