Part 1:

Golda Meir was well-known for pithy, powerful quotes. Among her most famous attributions (though not all are well-sourced) are quotes such as, "Peace will come when the Arabs will love their children more than they hate us", and "When peace comes we will perhaps in time be able to forgive the Arabs for killing our sons, but it will be harder for us to forgive them for having forced us to kill their sons". These types of quotes went over quite well for her persona and its public image, but as you've noted, they can be quite simplified. This quote in particular that you've asked about is one that has a rather interesting backstory.

To understand the quote, first you should probably understand Golda Meir's background, as you've mentioned. Meir was a member of Israel's founding generation, and a close associate and worker on behalf of Israel's most prominent leaders, from Ben-Gurion to Shimon Peres. She was a mountain in her own right, however, which led to comparisons between her and Thatcher (Meir is often referred to as Israel's "Iron Lady").

Golda Meir was born Golda Mabovitch in Russia in 1898, to a poor family and in a time of rising anti-Semitism and pogroms against Jews. In 1909, her family joined her father (who had fled 3 years earlier) in the United States, and she grew up a Zionist, believing in the necessity and propriety of a Jewish state quite ardently. She married, becoming Golda Myerson, and moved to what became the British Mandate in 1917. She worked her way up the political chain, eventually going to work as the Jewish Agency's Political Department's head. She was an effective speaker and an even more effective political operator. When war was raging in January 1948, Ben-Gurion sent her to the United States to fundraise, asking for $25 million. She raised $50 million instead. That money paid for arms shipments that helped turn the tide when the Arab invasion began in May 1948, after the civil war was all but over.

Meir was a diplomat as well as a political head and fundraiser. In 1947 and 1948, she held secret meetings with King Abdullah of Jordan to attempt to reach some kind of deal that would prevent his participation in a war, while in exchange Abdullah would take over the West Bank. While the pact was never kept, he met with Meir to warn her that war was coming, blaming pressure from Arab states and his populace. Meir served as Foreign Minister, and was known for laying down hard red lines dictating Israeli positions (for example, she stated in 1957 at the UN General Assembly that should the Egyptians close to the Straits of Tiran again, Israel would consider it an act of war and respond accordingly.

In short, Meir was a hardened woman, experienced in politics and diplomacy, and not prone to suffering fools, to put it that way. She had lived in the Middle East a long time, spoken with Arab and American and European leaders, and observed international conferences. She had come from very little and grown up to be Israel's Prime Minister following Israel's greatest success and at a time of almost unparalleled geopolitical strength in the Middle East (in terms of its position relative to Arab states, or how it felt at least).

So, now that we've understood a bit of her personality, what about the quote? The quote is one she sought to explain, such as this piece printed in the New York Times in 1976 (reprinted in Aish of course). The general argument in her quote is that, prior to the creation of an Israeli state, there was no Palestinian people defined as a nation of their own. To this end, she quotes a few individuals who made the same claim on behalf of Arab states or leagues. But she also seems to draw a distinction, one that views nationhood as tied to statehood in this context. During this period, there began a significant amount of agitation for a Palestinian state for the Palestinian people, exacerbated by the Israeli taking of the West Bank and Gaza in the 1967 war. Prior to that point, the Palestinian people had lived primarily as either refugees, or under Jordanian and Egyptian control and occupation. Now that Israel was in charge of the entire former British Mandate, Palestinian nationalism essentially kicked into overdrive, and began to gain traction (particularly since Fatah and the PLO had begun to attract attention, and Fatah eventually took control of the PLO by around this time). 1967 did more than merely awaken Palestinian nationalism, it also dampened the forces of pan-Arabism that Nasser had pushed significantly (and effectively) for a decade. Meir was of the belief, as were many Zionist leaders until a couple of decades later, that Palestinian nationalism was nothing more than a tool or prop to use to try and delegitimize the Jewish state. This of course was not helped by the fact that Palestinian nationalism was used by Arab leaders as a tool for their own benefit. It is notable after all that the PLO was originally founded as a tool for Egyptian use, after all, and that many Palestinian groups conducted terrorist attacks pre-1967 largely with the help of an Arab state sponsor. Following 1967, however, these groups became more independent, and Meir's quote was a reflection of the deep-rooted belief that the "other" nationalism did not truly exist. Given the political and social context of the conflict, this is not surprising; numerous studies and countless histories, including in the modern era, have sought to delegitimize the nationalism of an opponent when that nationalism conflicts with one's own. Indeed, Arab leaders and Palestinian Arabs long made this precise argument regarding Jews, saying that they did not constitute a nation at all. Indeed, the PLO Charter of 1968 says, in Article 20, "Nor do Jews constitute a single nation with an identity of its own; they are citizens of the states to which they belong." This was a theme echoed by Arab leaders for decades, and similarly, one can quote Jewish individuals supporting the view out of their own purposes (some feared anti-Semitism and accusations of dual loyalty, for example). So the context of this quote is that Meir believed Palestinian nationalism to be a political tool, not a real movement, and viewed Arab attempts to delegitimize Jewish nationhood as part of this approach. Ironically, the situation went both ways, of course, as Palestinians were no more willing to accept Jewish nationhood at this point.

Part 2:

As to your second question, there it gets a bit thornier. What constitutes a nation and identity are very nebulous, hard to define things, particularly when discussing how popular certain ideas are and what threshold that imposes for nationhood. The status of Palestine is easy: Palestine had never been a real, bounded state of its own. The area had been ruled by the Ottomans, Arab caliphates, and others, and while the name was used to refer to a particular area, it was not an administrative province or sovereign state under the Ottomans. It was divided (and redivided) into different provincial pieces, but when one referred to "Palestine", they almost always meant a geographical area that encompassed at least part of (if not more than) what is today Israel, the West Bank, and Gaza. The formal administration of "Palestine" as an entity began under the British, who set borders under the British Mandate in the 1920s. The text of the Mandate granted by the League of Nations made clear that "Palestine" did not have to end at the Jordan River, as it eventually did under British determinations. In Article 26 of the Mandate, it says, "In the territories lying between the Jordan and the eastern boundary of Palestine as ultimately determined", which already implies that the eastern boundary could (and perceptually, did) extend beyond the Jordan River. However, the British chose to use the Jordan River to mark the eastern boundary of "Palestine", and separated off Transjordan to the east ("Jordan" today). Following British control, there was Israel, Egyptian-occupied Gaza, and Jordanian-occupied-and-annexed West Bank. There was no status of "Palestine" as a state. In this, Meir is somewhat correct; they did not have sovereign control of a country. However, by the founding of Israel, it is pretty well-accepted that Palestinians had already formed a national identity, and considered themselves to be a people, at least in the majority. Some may have preferred pan-Arab identity, or pan-Islamism, but those were more a minority than anything else; those ideas weakened greatly in the 1920s and pan-Arabism made a comeback only after Israel's founding in earnest. While the ordinary "start" of nationhood or peoplehood for Palestinians is disputed (i.e. Khalidi claiming that "Palestinian-ness" began well before the 19th century or Kimmerling claiming national thought began around the early 1800s or others claiming it did not take shape until the 1880s or later), what constituted a "nation" was definitely formed by the period of 1948, despite the lack of a "Palestine" as a state. The complexity of a starting point is underlined by how many different identities Palestinians held, based on religion, Ottoman control, region, etc.. To make this more potent, imagine asking an American in Texas in the 1950s who they were. Their identity would be overridingly American, but they would also consider themselves a Texan, a Southerner, a member of some faith, a resident of their city, etc.. Now imagine that, but overlay on top of it the fact that civic identity was nowhere near as strong in the late 1800s and early 1900s point in the Ottoman Empire, so other forms of "nationhood" can start to take root as people search for a defining feature other than Ottomanism.

That debate aside, Meir's quote is both understandable, common, and reciprocal in Arab culture. While some continue to deny Palestinian peoplehood, and others continue to deny Jewish peoplehood, by the 1990s there was a definitive shift in many of these views. Following the overtures for peace made by both sides in the 1990s, including with the signing of the Oslo Accords (and obviously, this began before then, but this was a notable moment), there was a mutual recognition to some extent of peoplehood at the leadership level, which trickled down (though again, not fully) to the populaces. After all, the 1993 Oslo I Declaration of Principles included an exchange of letters, in which Yasser Arafat (head of the PLO) agreed to recognize the right of Israel to exist, and Yitzhak Rabin (Israel's Prime Minister) recognized Yasser Arafat as the representative of the Palestinian people specifically.