DID is one of the most controversial psychiatric disorders, with no clear consensus on diagnostic criteria or treatment. To answer your question on why MPD was renamed DID, for completeness, there is a bit of history to bear in mind and we will look at the versions of the DSM over the years and how things changed.
The DSM-II used the term Hysterical Neurosis, Dissociative Type. It described the possible occurrence of alterations in the patient's state of consciousness or identity, and included the symptoms of "amnesia, somnambulism, fugue, and multiple personality."
The DSM-III grouped the diagnosis with the other four major dissociative disorders using the term "multiple personality disorder".
The DSM-IV made more changes to DID than any other dissociative disorder, and renamed it DID.
The name was changed for two reasons.
The change emphasises the main problem is not a multitude of personalities, but rather a lack of a single, unified identity and an emphasis on "the identities as centers of information processing".
The term "personality" is used to refer to "characteristic patterns of thoughts, feelings, moods and behaviors of the whole individual", while for a patient with DID, the switches between identities and behavior patterns is the personality.
It is for this reason the DSM-IV-TR referred to "distinct identities or personality states" instead of personalities. The diagnostic criteria also changed to indicate that while the patient may name and personalise alters, they lack an independent, objective existence.
The DSM-IV-TR criteria for DID have been criticised for failing to capture the clinical complexity of DID, lacking usefulness in diagnosing individuals with DID (for instance, by focusing on the two least frequent and most subtle symptoms of DID) producing a high rate of false negatives and an excessive number of Dissociative disorder not otherwise specified (DDNOS) diagnoses, for excluding possession (seen as a cross-cultural form of DID), and for including only two "core" symptoms of DID (amnesia and self-alteration) while failing to discuss hallucinations, trance-like states, somatoform, depersonalisation, and derealisation symptoms. Arguments have been made for allowing diagnosis through the presence of some, but not all of the characteristics of DID rather than the current exclusive focus on the two least common and noticeable features.
The DSM-5 updated the definition of DID in 2013, and the summary of changes listed the changes as:
Several changes to the criteria for dissociative identity disorder
have been made in DSM-5. First, Criterion A has been expanded to
include certain possession-form phenomena and functional neurological
symptoms to account for more diverse presentations of the disorder.
Second, Criterion A now specifically states that transitions in
identity may be observable by others or self-reported. Third,
according to Criterion B, individuals with dissociative identity
disorder may have recurrent gaps in recall for everyday events, not
just for traumatic experiences. Other text modifications clarify the
nature and course of identity disruptions.