The Gemarah (36a) says the derash of Shmuel that לחם עוני is שעונין עליו דברים הרבה. Many words are said over the matzah. What are the many words said over the matzah? Rashi says the Haggadah and Hallel while Rabbenu Chananel says it refers to when we say מצה זו שאנו אוכלים על שום מה. These two interpretations are not just a machlokes as to what words are being referred to but reflect two different understandings of the derasha. One may say the derasha telling us a law in matzah, that the eating of matzah is incomplete unless it is accompanied by words and that is the view of the Rach, that when one eats matzah the explanation of matzah must be said (one can debate if this derash would actually mean that is one didn't say the explanation of matzah if one actually didn't fulfill matzah or is lacking in the optimal fulfillment of the mitzvah as the Ramban (Milchamos beginning of Berachot says.) Rashi on the other hand, understands that this is a halacha in סיפור יציאת מצרים. When one does not have the matzah present for the סיפור, then one is lacking in the סיפור.
As an outgrowth this affects the understanding of the statement of Rabban Gamlienl כל מי שלא אמר שלשה דברים בפסח לא יצא ידי חובתו - which chiuv? According to the Ramban following in the footsteps of the Rach, it refers to the obligation of matzah, maror and pesach. According to the Rambam (Chametz U'Matzah 7:5) it refers to the obligation of Haggadah for he does not hold that the matzah itself needs words attached to it.
These two approaches may be reflected in a difference in the Rishonim explaining why the matzah is brought back for the recitation of the Haggadah. The Tosfos (114a) say in order to say מצה זו על שום, (and the Bach 473 adds we are afraid we will forget to bring it back at that point in the Haggadah, so we do it at the beginning of the Haggadah.) In other words, it is only necessary to have the matzah present for the sake of the matzah but Rabbenu Dovid says it is in order for the matzah to be present during the sippur yitzias mitzraim.
The Rambam (8:6) cites the derasha of דרכו של עני בפרוסה and not דברים הרבה and in addition he has yachatz done right before the eating of the matzah, not before the Haggadah, indicating that he holds that the derasha of דברים הרבה is rejected before the derasha of פרוסה and there is no need to have any part of the Haggadah said before broken matzah. The Rambam (8:4) also does not hold of the matzah being present during the entirety of the Haggadah but has it brought back right before saying מצה זו. Our custom to do yachatz before the Haggadah and keep the broken piece on the table, the Alter Rebbe explains (473:36) is because we hold both derashot stand. It needs to be a piece of matzah and at the same time, the Haggadah must be said over it therefore we break it before starting the Haggadah. However, it is noteworthy that the Beis Yosef and the Ritva in the Haggadah say that we do yachatz before הא לחמא עניא since we are say הא לחמא עניא we want a live demonstration of the poor man's bread. Why do they not say because of the דין of דברים הרבה? In Yarach L'Moadaim siman 34 he cites his son and son-in-law suggest that the fulfillment of lechem עוני as a פרוסה is a din in the maaseh achilah of matzah that is has to be an eating of a pauper which is a פרוסה . However, the law of עוני meaning דברים הרבה needs only the actual cheftzah of מצה not the פרוסה.
The Gemarah (115b-116a) has additional derashot about לחם עוני that it is like the word עני to indicate that one uses a broken piece of matzah or that just as a pauper lights throws the bread in the oven fast before it rises, so too matzah can't rise. Those derashot tell us about the nature of the matzah so it is logical to assume the derasha of עונין עליו דברים הרבה also is about the matzah itself. However, the Michilta (cited in the Haggadah) says בעבור זה לא אמרתי אלא בשעה שיש מצה ומרור מנחם לפניך which is the possuk of the mitzvah of סיפור יציאת מתרים indicating that there is a law in sippur that the matzah must be there. This may be a source to say that matzah is needed for the fulfillment of sippur. One may in fact argue that these are two halachot. The halacha of דברים הרבה is a law in the matzah while the law in בשעה שיש מצה לפניך is a law in sippur. In the Briskor Haggadah it says the Briskor Rav explained the law of דברים הרבה is a law in sippur (and he compares it to kiddush said over wine,) and it concludes therefore, he was מקפיד that the matzah that was eaten be everyone should be one the table during the Haggadah. The "therefore" is difficult for if it is just a law in sippur, it should suffice with some matzah on the table, why does all the matzah that is eaten have to be there, that would indicate it is a law in matzah? It would seem that maybe the Briskor Rav held both elements are true, there is a law in sippur for the matzah to be present and there is a law in the matzah that is eaten that words must be recited over it (based upon בזמן שמצה ומרור מונחים לפניך', בגדרי מצות יחץ', there is room to be מפלפל in many of the points.)
R' Brown points out (Moadanay Moshe siman 23) that these two approaches are inversed regarding if the Haggadah must be said over the second cup of wine. The Alter Rebbe says (473:40) וְאַף עַל פִּי שֶׁאֵין צָרִיךְ לֶאֱחֹז הַכּוֹס בְּיָדוֹ עַד שֶׁיַּגִּיעַ לִ"לְפִיכָךְ" כְּמוֹ שֶׁיִּתְבָּאֵר, אַף עַל פִּי כֵן צָרִיךְ לִמְזֹג מִיָּד קֹדֶם הַתְחָלַת אֲמִירַת הַהַגָּדָה, כְּדֵי שֶׁיִּשְׁאֲלוּ הַתִּינוֹקוֹת This is based upon the Tur following the interpretation of Rashi and Rashban (116a.) In other words, there is no need to have a cup of wine present at the time of the recitation of the Haggadah, we only pour it in advance to encourage questions. However, the Rambam (7:3) says that the Haggadah is said over the second cup. In other words, part of the takkanah of the cup is that the Haggadah should be said over it. This is why the Rambam in the text of his Haggadah has the second cup poured already before the הא לחמא עניא.