The actor's name was Shakspere, as shown on his baptismal and burial records—and all other legal and family documents. Never in his lifetime was he called "Shakespeare" (not until 7 years after his death in the First Folio of 1623). Nor was he ever honored or acknowledged as "the soul of the age" in his lifetime. His death in 1616 passed without public notice. No one from the worlds of publishing or theater commented on or lamented his passing. Hardly the kind of send-off one would expect for the Star of Poets.
The statement "Shakespeare wasn't Shakespeare" creates a logical double-bind, the mind instinctively rejects such a proposition. However, it is far less confusing and more to the point to say "Shakspere wasn't 'Shakespeare'" if that is your belief.
Moreover, Marley had dibs on the name "Shakespeare"! Prior to his untimely murder at the age of 29 in 1593, he had written a long poem about Venus and Adonis. Following his 'death', it was published with no name on the title page, but with a dedicatory epistle that was signed "William Shakespeare" (He could hardly sign his own name.) A second narrative poem Rape of Lucrece was published the following year. As a result, until about 1598, the author Shakespeare, was known primarily as a premiere poet. Plays were secondary, published anonymously or not at all.The fact that most of the plays in the Shakespeare First Folio had not been previously printed attests to this.
In fact, the actor did not make his entrance as an author until the First Folio, when the legend was born—seven years after the Shakspere's death. And the Shakspere=Shakespeare myth was born.