As the question says, what are those three stages of Bhakti viz. Manda Bhakti, Madhyama Bhakti and Uttama Bhakti?
IMO what you're calling as Uttama Bhakti (best kind of devotion) can be called Sattvik Bhakti as well. What is Manda Bhakti (worst kind of devotion) in your language can also be called Tamasi Bhakti and etc.
The Devi Bhagavata Purana (DBP) has a discussion regarding these 3 kinds of such Bhaktis plus a 4th one which is the highest form of Bhakti (devotion).
- The Himâlayâs said :-- "O Mother! Now describe your Bhaki Yoga, by which ordinary men who have no dispassion get the knowledge of Brahma easily. 2-10. The Devî said:--"O Chief of Mountains! There are three paths, widely known, leading to the final liberation (Moksa). These are Karma Yoga, Jñâna Yoga and Bhakti Yoga. Of these three, Bhakti Yoga is the easiest in all respects; people can do it very well without incurring any suffering to the body, and bringing the mind to a perfect concentration. This Bhakti (devotion) again is of three kinds as the Gunas are three. His Bhakti is Tâmasî who worships Me, to pain others, being filled with vanity and jealousy and anger. That Bhakti is Râjâsic, when one worships Me for one's own welfare and does not intend to do harm to others. He has got some desire or end in view, some fame or to attain some objects of enjoyments and ignorantly, and thinking himself different from Me, worships Me with greatest devotion. Again that Bhakti is Sâttvikî when anybody worships Me to purify his sins, and offers to Me the result of all his Karmas, thinking that Jîva and Îs'vara are separate and knowing that this action of his is authorized in the Vedas and therefore must be observed. This Sâttvikî Bhakti is different from the Supreme Bhakti as the worshippers think Me separate; but it leads to the Supreme Bhakti. The other two Bhaktis do not lead to Parâ Bhakti (the Supreme Bhakti or the Highest unselfish Love.)
So, according to this passage, there are 3 kinds of basic Bhaktis, with the last one among which being capable of leading the devotee towards the highest form of Bhakti called the Para Bhakti.