You are correct in asking the question two different ways, in terms of ownership and beneficiaries. In each case, there is no harm in making the trust the owner and beneficiary, but it also may not be necessary. Changing ownership to the trust would give your successor trustee control of the trust and the ability to borrow against any cash value in the policy. However, you may never want to make any changes to the policy or borrow anything against it and, in any case, your attorney-in-fact under your durable power of attorney should be able to take these steps if necessary. So changing ownership probably serves little purpose.
Changing the beneficiaries to the trust may or may not serve a greater purpose depending on the terms of your trust. If your trust will continue after your death for the benefit of your spouse or children and you would want all funds you leave to be protected and managed in the same way, then the trust should be named beneficiary. On the other hand, if your trust will be distributed outright upon your death and the trust and the life insurance policy have the same beneficiary or beneficiaries, then changing the beneficiary to the trust provides little or no benefit.