Advance warning: 1. I am no expert on Hebrew, nor for that matter on semantics. 2. Comments relating to word usage are welcome below; attempts to refight the conflict are not.
One thing that's always puzzled me about pro-Israel discourse is the enormous weight it tends to place on the concept of nationhood. Over and over again, you find Zionists insisting that Israel is a nation, that Palestine is not a nation, that Palestine must acknowledge that Israel is a nation - as if nationhood were the key issue at stake. In pro-Palestine discourse, on the other hand, the question of whether either party is a nation hardly arises except in responses; who cares? Recently it struck me that this difference in rhetoric could perhaps be understood in semantic terms. What is a nation? And just how badly does this rather polysemous word translate?
When Netanyahu defends proposals to define Israel as the "nation-state of the Jewish people" (מדינת הלאום של העם היהודי), or declares that "we do not want a bi-national state" (איננו רוצים מדינה דו-לאומית), the word he's using is le'om לאום. In the Hebrew Bible, this word (better transcribed lə'ōm) is fairly uncommon, occurring 26 times in the plural and 5 in the singular (26 of them in the singular), usually in poetry and paired with its far more frequent near-synonym ʕām עָם "people, nation". It occurs in the singular in Proverbs, with general reference, and in Genesis 25, in which Jacob and Esau are each presented as a le'om, implicitly representing the nation made up of his descendants: respectively, Israel and Edom. Elsewhere, it occurs in the plural, especially referring to other nations (eg Psalms 47:3). In a more modern context, le'om is the word used for "ethnic affiliation" on Israeli ID cards: an Israeli citizen's le'om can be Jewish, Arab, or Druze, but, curiously enough, never Israeli.
In Arabic, the "nation-state of the Jewish people" is rendered as "دولة قومية للشعب اليهودي", and "bi-national state" as "دولة ثنائية القومية", both using the word qawmiyy "national", from qawm "nation, people". The latter word is very frequent in the Qur'an, occurring 383 times. Its usage there, however, is rather different. It never occurs pluralised (though it takes plural agreement). Whereas the only le'oms defined by name in the Bible are defined by common patrilineal ancestry, a qawm is defined by name in the Qur'an only in terms of their prophet or (more rarely) leader: qawmu Mūsā "the people of Moses", qawmu Firʕawn "the people of Pharaoh", qawmi Nūḥ "the people of Noah", qawmi 'Ibrāhīm "the people of Abraham"... Otherwise, it is defined in terms of its characteristics: most often, believing (aware, realising, grateful, etc.) or unbelieving (wrong-doing, misguided, self-wronging, etc.)
As might be expected based on this, the noun qawm is hardly ever used to refer to something like a "Palestinian nation", nor to a "Jewish nation", nor even to an "Arab nation" (Google returns derisorily small numbers of hits, in the low thousands or below). If you search for dawlah qawmiyyah "nation-state", what you mostly get is discussion of Israel (along with a few fringe movements). Qawmiyyah, "nationalism", is a more prominent concept in the modern era - above all, al-qawmiyyah al-ʕarabiyyah "Arab nationalism", ie the dream of a single pan-Arab state - but one with a rather ambivalent ring to it at best; there are still Arab nationalists around, but the Arab unity project has had a musty 1960s smell to it for a while, criticised as much from the right as from the left. Even apart from its content, the common Israeli demand for recognition of a "nation-state of the Jewish people" thus translates rather poorly - not because the concepts don't exist, but because they don't have similar connotations. Palestinians aren't normally speaking in terms of a "dawlah qawmiyyah of the Palestinian people", nor for that matter of "the Arab people"; whether Palestine is a nation-state or some other kind of state is a secondary issue.
When Mahmoud Abbas mentions "national institutions" or a "national unity government" in his UNGA speech - or meets with the Palestinian National Council - the word he's using, and the word any Arabic speaker would use, is waṭaniyy وطني, from waṭan وطن "nation, homeland". In the Palestinian declaration of independence of 1988, qawmiyy occurs once (in a token nod to Arab nationalism), whereas waṭaniyy occurs 13 times, in collocations like "national identity", "national independence", "national rights", "national personality", "national will"... The word waṭan doesn't occur in the Qur'an at all; the closest it comes is a single usage of mawāṭin "regions". It unambiguously refers to a land, not to a group of people. And, unlike qawm, it has a profound resonance in the context of Palestinian - and Arab - nationalism, and not just because it provides the adjective used in collocations like "national liberation" or "national anthem". It recurs nostalgically in the poems of Mahmoud Darwish ("What is the waṭan? It is the house, and the mulberry tree, and the chicken coop and the beehive, and the smell of bread and the sky") or Tawfiq Zayyad ("As you were, so you shall remain, my waṭan - present in the leaves of the oleander and the fragrance of jasmine"). Further afield, Nizar Qabbani's remarkable line comes to mind now more than ever: "O my waṭan, have they made you a serial of horror whose events we follow in the evening? Then how shall we see you if they cut the power?" And, of course, waṭaniyyah وطنية "patriotism" has far more positive connotations than qawmiyyah. All of this vocabulary places the emphasis away from notions of ethnic cohesion or common ancestry, focusing on a different common ground: the land itself.
The Hebrew translation of waṭan would appear to be moledet (מולדת) "homeland". This word does occur in the Bible, 22 times - but, like le'om, it carries there a sense much more closely tied to human kinship, referring to "kindred, family" as well as "birthplace, native country". Israel's declaration of independence refers to the land as the "national homeland (moledet)" of the Jews, and there seems to be a good deal of Hebrew poetry on the subject. However, collocations like "national anthem" or "National Council" or "national unity government" or "national liberation" all derive from le'om, not from moledet.
"Nation" in English usage is ambiguous: is a nation united primarily by its attachment to a given area, or by its common ancestry? Either language can express either idea. However, the best-established and most positively viewed terminology in Arabic focuses on the former, while in Hebrew it focuses on the latter. This difference is hardly the source of the conflict, but it does play some small part in impeding mutual understanding.