Final answer:
The poem by Browning, refusing to reveal personal life details, is a dramatic monologue, where a created persona speaks, distinct from the poet.
Step-by-step explanation:
The Browning poem where the poet refuses to reveal personal details about his own life is best described as a dramatic monologue. This literary form involves a single character, who is not the poet, speaking to a silent audience. Robert Browning is renowned for his dramatic monologues, such as "My Last Duchess", wherein characters often reveal their personalities and situations through their speech. The key feature of this form is the creation of a persona that is distinct from the poet, which allows the poet to explore different voices and perspectives. Charlotte Mew's poem and the Robert Browning poems mentioned in the information provided serve as examples of this form of poetry.