Are islam and iman one and the same?

In "the proof that faith increases and decreases" Abu Dawud relates a report in which the Prophet distinguishes the description "believer" from "Muslim", and al-Khattabi pauses on it to say, "How often people go wrong on this question!" ²

Al-Zuhri relates from 'Amir ibn Sa'd, from his father Sa'd ibn Abi Waqqas, that the Prophet gave to some men and left one of them out; Sa'd said, "You gave so-and-so and so-and-so and did not give so-and-so, though he is a believer," and he answered, "Or a Muslim," until Sa'd had repeated it three times; then he said, "I give to some men and leave one dearer to me than they, giving him nothing, for fear that they be thrown on their faces into the Fire." ¹ The annotator, quoting Muhammad Muhyi al-Din 'Abd al-Hamid, explains "or" here as a correction — say rather that he is a Muslim, and do not assert his faith, since what hearts conceal is known only to God. ³

Ma'mar relates from al-Zuhri on al-Hujurat 14: "We hold that islam is the word and iman the deed." ³

Al-Khattabi says that al-Zuhri held what Ma'mar reports, citing the verse, while others held that faith and islam are one thing, citing al-Dhariyat 35–36: God promised to bring the believers out of Lot's people, then said He found there only one household of Muslims, so that islam indicates faith. ² He notes that two leading scholars discussed this, each adopting one view, and that the later one wrote against the earlier a book of some two hundred leaves. ²

Al-Khattabi names neither the two scholars who disputed the question nor the book; the views of the theologians and the Maturidis on the difference between the two names are not given here.