Given a symmetric monoidal category one can construct it's underlying operad (well, symmetric colored operad, but I won't keep mentioning this). This operad has the same objects as the SMC. The operads arising this way can be characterized, so that we can regard a SMC as a special kind of operad.

Conversely, given an operad, one can construct its symmetric monoidal envelope, a SMC monoidal category whose objects are not just the objects of the operad but rather formal (commutative) products of them. The SMCs arising this way can be characterized, as David mentions in the question, so that we can also regard operads as special SMCs.

Now, when operads and SMC are implemented by Lurie's definitions, the direction in which it is easy to go is that of regarding a SMC as an operad. If the SMC is given by a functor $p : C \to Fin_{\ast}$, then regarded as an operad it is the same functor, and whether we view it as a SMC or as an operad its objects are the objects of the category $p^{-1}\langle 1 \rangle$.

To go in the other direction, we need to change the functor: given an operad $p : C \to Fin_{\ast}$, its symmetric monoidal envelope will be a different functor $q : D \to Fin_{\ast}$. The objects of the envelope, which are the objects of $q^{-1}\langle 1 \rangle$, will be all the objects of $C$, not just the objects of the subcategory $p^{-1}\langle 1 \rangle$.

Lurie constructs the symmetric monoidal envelope in section 2.2.4 of Higher Algebra.

Given a symmetric monoidal category one can construct its underlying operad (well, symmetric colored operad, but I won't keep mentioning this). This operad has the same objects as the SMC. The operads arising this way can be characterized, so that we can regard a SMC as a special kind of operad.

Conversely, given an operad, one can construct its symmetric monoidal envelope, a symmetric monoidal category whose objects are not just the objects of the operad but rather formal (commutative) products of them. The SMCs arising this way can be characterized, as David mentions in the question, so that we can also regard operads as special SMCs.

Now, when operads and SMC are implemented by Lurie's definitions, the direction in which it is easy to go is that of regarding a SMC as an operad. If the SMC is given by a functor $p : C \to Fin_{\ast}$, then regarded as an operad it is the same functor, and whether we view it as a SMC or as an operad its objects are the objects of the category $p^{-1}\langle 1 \rangle$.

To go in the other direction, we need to change the functor: given an operad $p : C \to Fin_{\ast}$, its symmetric monoidal envelope will be a different functor $q : D \to Fin_{\ast}$. The objects of the envelope, which are the objects of $q^{-1}\langle 1 \rangle$, will be all the objects of $C$, not just the objects of the subcategory $p^{-1}\langle 1 \rangle$.

Lurie constructs the symmetric monoidal envelope in section 2.2.4 of Higher Algebra.