Every trust has different parties that have different rights and obligations based on their role as expressed in the trust instrument. The settlors are the people who are signing to set up the trust. The settlors may also have other roles, but their only role may be as the settlors of the person who set up the trust.
According to Policy Genius, the person who is in charge of managing and following the terms of the trust is called the trustee. Normally, in your typical avoid probate trust set up by a couple, that couple are the first co-trustees. Then perhaps when one of them dies, the surviving spouse is the sole trustee. After both partners die, maybe their daughter is the successor trustee to be in charge. There may even be another backup named after their daughter. After one of them dies, the surviving spouse would be the trustee to handle a home sale.
Then you have beneficiaries to a trust. According to The Balance, there are many types of beneficiaries. There are income beneficiaries and principal beneficiaries. Income beneficiaries are people who are entitled to the trust income. So if a couple sets up their revocable living trust, they are likely to be the income beneficiaries. If their stocks and bonds were in their trust, that income goes to the spouse because they are the income beneficiaries. They are entitled to receive the trust income.
Typically, at a later date, the principal beneficiaries are entitled to receive the trust assets. So maybe in your typical avoid probate living trust, the children may be the principal beneficiaries. When the parents die, then the children are entitled to the trust principal.