At least in the United States, you are correct to point out a relationship between former slave states and birthright citizenship laws. However, birthright citizenship actually was introduced as a way to easily incorporate formerly enslaved black people into the political process. I can go into detail about the American legal history here, but I know little about the legal histories of Spanish colonial states, so other nations on the map likely have similar but unique paths to birthright citizenship.

Martha S. Jones has a fantastic book about the push for birthright citizenship called Birthright Citizens: A History of Race and Rights in Antebellum America. She particularly focuses on the push for birthright citizenship in free black community in antebellum Baltimore. I highly recommend this book to anyone who is interested in this topic because it is relatively short, very engaging, and centers the stories of free black communities who often get forgotten about in the antebellum period. Most of my argument is simply a summary of her work, so I recommend you check it out. Essentially, her argument is that the United States did not quite know what to do about the freed black population in the antebellum period. Some legal rights, such as the right to sue in court, had been extended to them, but they lacked the stable legal position offered by citizenship. Free black people were extremely at risk of being mistaken for a fugitive slave and being carried off to the South again, despite many black people in the Baltimore population having been born free. This risk was a large source of anxiety for these communities, and the safety and protection that came with birthright citizenship was very attractive. While they could become naturalized citizens, and many did, the process was seen to be unfair -- after all, many of them had been born free just like white people.

Towards the middle of the 19th century, discussion about slavery had begun to heat up. Around the globe, European powers were beginning to cut their ties to the slave trade and abolish it within their empires. The Spanish had already abolished slavery in the 18th century, and the English outlawed it in their empire in 1834. The United States couldn't not think about the future of slavery in its borders. Abolition was gaining more traction, as were early women's rights movements that supported abolition. Many white abolitionists however, shared Thomas Jefferson's opinion that the deep animosity as a result of racial oppression meant that the races could not live together without "convulsions, which will probably never end but in the extermination of the one or the other race." This gave rise to the colonization movements that sought to end slavery by deporting black people back to Africa.

Understandably, this kind of talk made free black community leaders very nervous. Their legal status on the mainland was very weak and they had no legal claim to resist deportation, since they were not citizens. The Fugitive Slave Act of 1850 caused more problems for the black community. The crackdown on catching fugitive slaves put free black people in much more danger as well. This anxiety about the future created more outspoken activism for birthright citizenship. Additionally, birthright citizenship was seen as a way to subtly undermine the institution of slavery itself. If anyone born on US soil was automatically afforded the rights citizen, then slaveowners could not violate the rights of slaves born on US soil by keeping them enslaved. From here, we know the story -- tensions over slavery spiraled into civil war, the Emancipation Declaration was signed, and after the war, Americans had to put new thought into how to incorporate black people into the union as free individuals.

The 14th Amendment, passed on the heels of the Civil War in 1868 finally made birthright citizenship a legal reality. It states that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." It was passed in response to a need to decide whether or not freed slaves were citizens -- if black people are no longer property, then what exactly is their legal status? The conversations that had been happening in abolitionist circles were brought to the forefront to answer this question. Birthright citizenship was the product of decades of antebellum black activism, won only after slavery itself had been abolished. Abolition made the determination easy -- anyone who was born on American soil, whether they were born free or slave, was now a citizen of the United States. This reassurance brought stability to vulnerable free black populations and probably brought peace of mind to many white people who were made uneasy by the thought of newly freed black people with no national allegiance wandering the country. It was seen as a way to smooth over the divisions of the past and move towards a cleaner post-slavery future. It was likely for these reasons as well that birthright citizenship was implemented in states that were also trying to heal some of the social wounds of slavery.

TLDR: Birthright citizenship in the United States was instated in order to incorporate newly free black people into the legal system, providing them stability and documentation after slavery even if equality before the law was still a long ways off.

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Bibliography:

Martha S. Jones, Birthright Citizens: A History of Race and Rights in Antebellum America, 2018.

Thomas Jefferson, "Notes on the State of Virginia," 1785.

United States Constitution, 14th Amendment, 1868.

Just to answer for Canada's particular context - Canada inherited this practice from the British and remained linked to the status of "British subject" into the 1940s. Common law, which also applied to places like New Zealand and Australia, stated that anyone born in the British Empire was a subject of the British Crown. You could also become a subject by living there for three years then applying for naturalization. Being a subject could enable advantages like voting, if you met the voting requirements of the moment (which mostly involved property and maleness - but not always, women with property actually voted in New Brunswick until the provincial legislature realized what was going on and closed the loophole by specifying gender).

The first law of the Canadian Parliament referencing "Canadian citizenship" was made in 1910, and kept similar rules: if you're born in Canada and never took another citizenship, if you were a British subject who had lived here for at least three years, or if you were naturalized by a legal process, you counted as a "citizen". This definition of citizen was simple enough to keep track in an environment with nascent bureaucracy and a dispersed population, and to keep up in a country with high immigration rates.

In particular, Canada had large amounts of immigration from other parts of the British Empire, so letting them become citizens quickly encouraged that flow. In 1910 the government of Canada was pushing hard to colonize the prairies, and there were many racial and ethnic prejudices about non-British immigrants. To further assist the flow of fellow white English-speakers to Canada, the Canadian government adopted the same rules about naturalization as the rest of the British Empire in 1914.

Canadian citizenship was finally created as an entirely separate status from being a British subject on January 1, 1947. The separation occurred in an era when the British empire was becoming a bit smaller, and there were long processes regarding the independence of the Dominions: Canadian, Australia, New Zealand, and South Africa. These issues were often slowly negotiated in Commonwealth conferences (if you are insane and thus interested in this process, I'd recommend C.P. Stacey's "Canada and the Age of Conflict" which is a very factual recounting of most of Canada's foreign affairs, including all these negotiations of sovereignty and responsibility with Britain). A major turning point was the Westminster Statute of 1931, which removed Britain's authority to legislate for the Dominions. It's often written that Canada got more confidence as an independent country throughout the two world wars, but you could equally say that Britain was turning its focus inwards and less interested in babysitting Dominions (unless of course they needed those Dominions' help in a war).

The Act of 1947 transitioned all British subjects born in Canada or who had lived there more than 5 years, amongst others, into "Canadian citizenship" and specified that anyone born there from 1947 on would be a Canadian citizen, along with anyone who lived there five years and got naturalized, and a few others. Again, high rates of immigration encouraged citizenship by birth. You could lose citizenship by living away for more than 5 years or being naturalized into another country, meaning that the "edge cases" of people who were born there without their parents intending to stay would likely eventually give up citizenship anyway.

One quick correction:

It's more complicated than this, really. (your graphic /map is a bit simplifying.)

For instance:

France has long had citizenship based on territory (jus soli), which means being born in the area is crucial for determining citizenship. (i.e. citizenship accords to children that are born in France of foreign parents and that grow up in France.)

Of course, you have to reside in France to claim it, which does indeed seem to make it more restrictive than the USA, for instance.

Germany, on the other hand, for a century, had citizenship based on descent. (jus sanguinis, through "blood", i.e. children acquire at birth the citizenship of their parents.) This is a very different legal and conceptual notion. Indeed, this legal notion was so pervasive that in the Germany of the late 1990s, people of Turkish origin who had lived in Germany for 3 or more generations in Germany (and spoke only German) still could not become citizens, while a Ukrainian with a grandparent who had been in the SS (and thus, by German law of the 1940s, averred thereby to be "ethnically" German) could use these SS identity documents to claim German citizenship without ever having been to Germany or knowing a single word of the language. (!)

The classic work here is Rogers Brubaker's Citizenship and Nationhood in France and Germany, published in the early '90s, which explored the historical roots behind these two very different models of citizenship... and by extension, explored what a "nation" actually is...i.e. is it a territorial area, or a group of people who have something in common?

Brubaker had many interesting arguments as to why this difference (between these two nations that were neighbors) arose and became so different from each other. There's path-dependency, and many different factors at work.

But the one of his arguments that stuck with me was that in Germany, state-building preceded nation-building... namely, many German principalities (before the unification of Germany in 1870) had non-contiguous territory with porous borders, i.e. many spots on a map (rather than just one)... but still developed extensive bureaucracies (for taxation, law, etc.)... so that it became very important to apportion people "correctly," and place them in the proper jurisdiction... and to do so before territorial (i.e. "national") borders became hard-and-fast. (i.e. before the unification of the German lands in 1871.) But I'm not doing justice to the full nuance of his interesting argument.

When Brubaker's book came out, France and Germany contrasted starkly... but since then, they've moved closer together, with Germany (circa 2000, under pressure from the Green Party) moving away from jus sanguinis to include birth in Germany as a qualification for citizenship--Turkish Germans now have a path to citizenship--and with France (giving political unease about immigration from North Africa) adding qualifications to their jus soli (in the 1993 Méhaignerie Law) by requiring that citizenship be officially requested by a legal adult. (i.e. citizenship is not automatically granted, it must be officially requested... which can be a barrier.)

So, this is just to say: citizenship is more complicated than your map shows, and different conceptions of citizenship are highly structured by both culture and by paths of historical development.