Question
Is Mahishasura Mardini Lakshmi or Durga?
Answer
Godesses Ambika who is also known by the name Godesses Durga is the Mahisasura Mardini (slayer of Mahisasura). Precisely speaking it is Chandika form of Godesses Ambika who slays Mahisasura:
Chaptet 81 to Chapter 93 of Markandeya Purana is called Devi Mahatmyam and it sings the Glory of Devi.
First of all Mahisasura defeats all gods and take their position as described in chapter 82:
देवासुरमभूद्युद्धं पूर्णमब्दशतं पुरा।
महिषेऽसुराणामधिपे देवानां च पुरन्दरे॥२॥
तत्रासुरैर्महावीर्यैर्देवसैन्यं पराजितम्।
जित्वा च सकलान् देवानिन्द्रोऽभून्महिषासुरः॥३॥
In former times war was waged between the deities and demons for the full space of one hundred years; also between Mahisha-sura, the greatest of demons, and Purandara (Indra). The Asura of ample heroism repulsed the army of the Devas; Mahishasura vanquished all the gods and became as Indra.
Then in the same chapter Gods go to Brahma and pray him to give him relief from Mahisasura. Then from the combined energy of Shiva, Brahma, Vishnu, Indra and etc... all Gods Godesses Ambika/ Durga manifest as:
इत्थं निशम्य देवानां वचांसि मधुसूदनः।
चकार कोपं शम्भुश्च भ्रुकुटीकुटिलाननौ॥९॥
ततोऽतिकोपपूर्णस्य चक्रिणो वदनात्ततः।
निश्चक्राम महत्तेजो ब्रह्मणः शंकरस्य च॥१०॥
अन्येषां चैव देवानां शक्रादीनां शरीरतः।
निर्गतं सुमहत्तेजस्तच्चैक्यं समगच्छत॥११॥
अतीव तेजसः कूटं ज्वलन्तमिव पर्वतम्।
ददृशुस्ते सुरास्तत्र ज्वालाव्याप्तदिगन्तरम्॥१२॥
अतुलं तत्र तत्तेजः सर्वदेवशरीरजम्।
एकस्थं तदभून्नारी व्याप्तलोकत्रयं त्विषा॥१३॥
Madhusudana hearkened to the words of the deities; Sambhu was wroth contracting his brows with furious ire. Afterwards, Chakrini's face was transported with rage, abundance of light issued from Bramha and Sankara. An immense mass of light proceeded from the bodies of Sakra and all the other Devas, and conglomerated. This effulgent lustre appeared as a flaming mountain: the Suras having seen the flame receded to the extremities of regions. The peerless light issued from the bodies of all the Devas and the phenomenon was transfigured as a woman extending through the three worlds.
Then all Gods give their respective weapons, Himavat gives Lion and so on.. Then Godesses Ambika marches to slay Mahisasura. Then the slaying of Mahisasura is described in chapter 83:
एवं संक्षीयमाणे तु स्वसैन्ये महिषासुरः।
माहिषेण स्वरूपेण त्रासयामास तान् गणान्॥२१॥
कांश्चित्तुण्डप्रहारेण खुरक्षेपैस्तथापरान्।
लाङ्गूलताडितांश्चान्याञ्छृङ्गाभ्यां च विदारितान्॥२२॥
वेगेन कांश्चिदपरान्नादेन भ्रमणेन च।
निःश्वासपवनेनान्यान् पातयामास भूतले॥२३॥
निपात्य प्रमथानीकमभ्यधावत सोऽसुरः।
सिंहं हन्तुं महादेव्याः कोपं चक्रे ततोऽम्बिका॥२४॥
सोऽपि कोपान्महावीर्यः खुरक्षुण्णमहीतलः।
श्रृङ्गाभ्यां पर्वतानुच्चांश्चिक्षेप च ननाद च॥२५॥
वेगभ्रमणविक्षुण्णा मही तस्य व्यशीर्यत।
लाङ्गूलेनाहतश्चाब्धिः प्लावयामास सर्वतः॥२६॥
धुतश्रृङ्गविभिन्नाश्च खण्डं* खण्डं ययुर्घनाः।
श्वासानिलास्ताः शतशो निपेतुर्नभसोऽचलाः॥२७॥
इति क्रोधसमाध्मातमापतन्तं महासुरम्।
दृष्ट्वा सा चण्डिका कोपं तद्वधाय तदाकरोत्॥२८॥
सा क्षिप्त्वा तस्य वै पाशं तं बबन्ध महासुरम्।
तत्याज माहिषं रूपं सोऽपि बद्धो महामृधे॥२९॥
ततः सिंहोऽभवत्सद्यो यावत्तस्याम्बिका शिरः।
छिनत्ति तावत्पुरुषः खड्गपाणिरदृश्यत॥३०॥
तत एवाशु पुरुषं देवी चिच्छेद सायकैः।
तं खड्गचर्मणा सार्धं ततः सोऽभून्महागजः॥३१॥
करेण च महासिंहं तं चकर्ष जगर्ज च।
कर्षतस्तु करं देवी खड्गेन निरकृन्तत॥३२॥
ततो महासुरो भूयो माहिषं वपुरास्थितः।
तथैव क्षोभयामास त्रैलोक्यं सचराचरम्॥३३॥
ततः क्रुद्धा जगन्माता चण्डिका पानमुत्तमम्।
पपौ पुनः पुनश्चैव जहासारुणलोचना॥३४॥
ननर्द चासुरः सोऽपि बलवीर्यमदोद्धतः।
विषाणाभ्यां च चिक्षेप चण्डिकां प्रति भूधरान्॥३५॥
सा च तान् प्रहितांस्तेन चूर्णयन्ती शरोत्करैः।
उवाच तं मदोद्धूतमुखरागाकुलाक्षरम्॥३६॥
While she was thus destroying his army, Mahishasura impeded those Ganas in the form of Mahisha. Some by biting with his mouth, and others by kicking with his hoofs, some by the lashes of his tail and by the goring of his horns. Some of them by the loudness of his bellow, by his burning breath he made them to fall to the ground. The Asura having caused Pramadas to fall, angrily ran to kill the lion of the goddess. The great hero enraged, tore up the earth with his hoofs and cast down the highest hills. He speedily turned about, cracked the earth, lashed the ocean with his tail, and made it to overflow everywhere. The clouds were dispersed by his long horns, the mountains and sky were blown into hundred pieces by his breath and sighs. The great Asura wrathfully bellowed! Chandika looked, highly enraged and endeavoured to lay hold on him. She threw out a Pasa to bind him, he was tied, and relinquished his buffaloe's form in this great combat. He instantly turned into a lion, within the time that Ambika cut of this head; he then converted himself into a man, held a sword and disappeared. Afterwards the goddess pierced him with her arrows, he with his sword and shield turned into a mighty elephant. The elephant pulled the lion with his trunk, and made a terrific noise, the goddess cut his proboscis with her sword. Afterwards the great Asura assumed the form of a buffaloe, and terrified the three Lokas of animates and inanimates. The passionate Chandika, matron of the world, frequently drank excellent liquor, bearing wrathful redness in her eyes. The Asura being highly endowed with bravery and lust, furiously hurled mountains with his horns on Chandika. She pulverized them with her strong arrows, and with increased intoxication and angry looking face.
देव्युवाच॥३७॥
गर्ज गर्ज क्षणं मूढ मधु यावत्पिबाम्यहम्।
मया त्वयि हतेऽत्रैव गर्जिष्यन्त्याशु देवताः॥३८
Devi cried "Oh! vain noise! noise! wait a moment until I perfectly drink this madhu; when you are slain by me here, all the gods will exclaim."
ऋषिरुवाच॥३९॥
एवमुक्त्वा समुत्पत्य साऽऽरूढा तं महासुरम्।
पादेनाक्रम्य कण्ठे च शूलेनैनमताडयत्॥४०॥
ततः सोऽपि पदाऽऽक्रान्तस्तया निजमुखात्ततः।
अर्धनिष्क्रान्त एवासीद् देव्या वीर्येण संवृतः॥४१॥
अर्धनिष्क्रान्त एवासौ युध्यमानो महासुरः।
तया महासिना देव्या शिरश्छित्त्वा निपातितः॥४२॥
ततो हाहाकृतं सर्वं दैत्यसैन्यं ननाश तत्।
प्रहर्षं च परं जग्मुः सकला देवतागणाः॥४३॥
तुष्टुवुस्तां सुरा देवीं सह दिव्यैर्महर्षिभिः।
जगुर्गन्धर्वपतयो ननृतुश्चाप्सरोगणाः॥
The Rishi said, she laid hold on the great Asura, (who advanced towards her conveyance), with her feet, she struck him with her Sula. When he was caught hold on by her feet, his half real form from his face became manifest; the goddess was surrounded by a blaze of glory. The great Asura being half issued and fighting with her, the goddess cut off his head with her ample sword. Afterwards every one exclaimed, that the army of the Daitya was annihilated; then a multitude of deities were highly delighted. The band of deities, with the magnificient sages applauded the goddess, the kings of Gandharvas sung, Apsaras danced.
Thus from above description, it is clear it is Godesses Chandika (a form of Ambika), who slayed Mahisasura. Then, Gods pray with Godesses Ambika in chapter 84, after she slays Mahisasura. There is a beautiful long prayer. The prayer goes as:
देव्या यया ततमिदं जगदात्मशक्त्या
निश्शेषदेवगणशक्तिसमूहमूर्त्या।
तामम्बिकामखिलदेवमहर्षिपूज्यां
भक्त्या नताः स्म विदधातु शुभानि सा नः॥३॥
O Goddess! by thy power this world was created with spirits and gods of different and numerous forms. O Ambika! thou art revered by all the deities; the magnificent sages faithfully prostrate themselves to thee: mayest thou give us prosperity.
As a Side note, from the chapter 85, story of Sumbha and Nisumbha begins, and at that time there is no need for Gods, to manifest Ambika from their powers. At this time Ambika directly come from body of Parvati and goes to slay Sumbha, Nisumbha and other Daityas:
एवं स्तवादियुक्तानां देवानां तत्र पार्वती।
स्नातुमभ्याययौ तोये जाह्नव्या नृपनन्दन॥८४॥
साब्रवीत्तान् सुरान् सुभ्रूर्भवद्भिः स्तूयतेऽत्र का।
शरीरकोशतश्चास्याः समुद्भूताब्रवीच्छिवा॥८५॥
स्तोत्रं ममैतत् क्रियते शुम्भदैत्यनिराकृतैः।
देवैः समेतैः समरे निशुम्भेन पराजितैः॥८६॥
शरीर कोशाद्यत्तस्याः पार्वत्या निःसृताम्बिका।
कौशिकीति समस्तेषु ततो लोकेषु गीयते॥
The Rishi said, when the deities applauded Parvati, O prince! she went to perform ablution in the water of Janhavi. She asked them, "Whom did you greet there." The resplendent Shiva emanated from her body, and spoke thus to her. "They recited my praises, because they were expelled by the giant Sumbha, all the deities retreated before Nisumbha in the war." From the body of Parvati, Ambika issued, and was denominated Kouciki throughout the universe. When she had emanated from Parvati, (who turned black), she was called Kouciki and resorted about Himachala.
What I speculate is at the time of Mahisasura slaying event, Parvati was not born yet (she was born only after Chakshusha Manvanatara, ) while Sati already died in Pitri Kalpa and there was no physical form of Shakti whom Devas could evoke to kill Mahisasura. So, it was necessary that all use their Shakti and manifest from there (as Shakti is pervaded in power of all), and thus Ambika/Durga issued from there at that time. While in case of Sumbha and Nisumbha, Parvati was already there and Devas simply go and pray to Parvati and then Godesses Ambika manifested from Parvati.
Edit Update:
As mentioned in question Devi Bhagvatam quotes:
This Lady was Mahâ Laksmî; composed of the three qualities of the three colours, beautiful, and fascinating to the universe.
MahaLakshmi mentioned here is not consort of Lord Vishnu. I explain it in my answer here. Quoting from that answer:
ततोऽम्बिकां परां शक्तिं महालक्ष्मीस्वरुपिणीम् । तुष्टुवुस्ते सुराः सर्वे भक्तिगद्गदया गिरा ।।
The Gods then prayed to the Supreme Godesses Ambika in the form of MahaLakshmi with choked voice in devotion. .
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एवं ते कथितो राजन्महालक्ष्म्याः समुद्भवः । सरस्वत्यास्त्तथोत्त्पत्तिं श्रृणु सुस्थेन चेतसा ।।O King in this way the origin of MahaLakshmi has been spoken to you. Now with a cool mind listen to origin of (Maha)Saraswati.
So, MahaLakshmi described here is just form of Godesses Ambika and not consort of Lord Vishnu.
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