[Drew68]

In this case it was referring to a specific president. Or is there more than one in office currently?

Doesn't work that way.

The rules are clear. "The president laid out his proposals." "The proposals laid out by President Trump specified..." "According to Senator Cruz..." "The senator spoke yesterday with reporters."

[TigersEye]

"The president laid out his proposals."

Is that not referring to a "specific" president or was your definition of grammatical terms incorrect?