For context as to Trotsky and Lenin’s overall ideological basis and approach during the events leading to the creation of the Soviet Union his own book The History of the Russian Revolution is helpful.

To address this specific question Trotsky’s other book The Revolution Betrayed is illuminating.

To explain in very broad strokes, it was not simply a matter of “wanting” to expand socialism or not, Trotsky (and Lenin) fundamentally believed that socialism could not exist if it did not spread to other countries.

First of all, the threat socialism posed to capitalism would mean that any socialist country would be ostracized and opposed at every opportunity until it eventually failed. Indeed, the Russian Civil War that followed the Revolution was heavily encouraged and supplied by other countries, over twenty of which went so far as to send actual troops into Russia to fight on the side of the Whites.

Throughout the Cold War period, nominally “socialist” or merely left-leaning states were repeatedly embargoed and were threatened by western meddling in their internal politics or even outright military intervention.

Second, an important part of Marxist economics is the rough idea of stages of development. Socialism, capitalism, feudalism, etc are defined by the method of production of the time and the social relationships it engenders, which is to say that certain technologies and advancements are required to have a certain economic system.

One could not go back in time and create a socialist society in Ancient Rome, the system of production and distribution simply does not allow for it. Just as capitalism could not exist in Han Dynasty China.

As a result, socialism requires capitalism to develop the means of production to a certain level before socialism becomes viable. This is why Marx (and even Lenin initially) believed that a socialist Revolution would happen in one of the advanced capitalist nations, not a relatively undeveloped backwater like Russia.

That doesn’t mean to simply ignore developments in poorer countries though, and so of course Lenin ultimately felt that the course of events meant that they had to seize whatever opportunity they had.

However, the undeveloped conditions of Russia, and the deaths of many of the most educated sectors of society in the Civil War, meant that the task of building socialism was incredibly difficult in Russia. Simply put they needed highly skilled workers and advanced manufacturing capabilities.

A socialist society is a post-scarcity society. Scarcity leads to class divisions of those who have resources and those who don’t. A hierarchy and conflict between classes is inevitable if there are not enough resources to go around, and it is only through the advanced organizations and technology that capitalism produces that socialism can then be viable as a successor.

Additionally, this is only possible through the complicated global supply chains we have become accustomed to. One single country, no matter how large, will always lack something it needs. Even relatively simply products are often made form minerals mined in half a dozen countries, refined in a few others, assembled in one more, and shipped for sale somewhere else. One single country, even one as large as Russia could not hope to provide all its citizens’ needs without significant trade with the capitalists, which would itself introduce capitalist dynamics into the country and risk a slow devolution into capitalism.

Even non-communist countries are clearly failing due to ostracism from the world at large. North Korea and Iran, even were their leaders interested in the welfare of their people, are simply incapable of providing a high standard of living due to their separation from the global market

Therefore, Trotsky believed that without expanding the Revolution, and in particular without expanding to the richer more developed countries of the west, that socialism would ultimately flounder and stagnate.

In his 1930 book (published in Germany due to the power struggles ongoing) Permanent Revolution Trotsky explains these ideas in more detail.

To address Stalin, his principal concern was with the consolidation, maintenance, and expansion of his own personal power. Due to the aforementioned issues of developing socialism in Russia, as well as the deaths of so many skilled and educated people, Stalin found himself as the emergent leader of a class of bureaucrats who were becoming entrenched as a power in their own right, and whose interests were rapidly diverging from those espoused by Lenin and Trotsky.

He wasn’t necessarily opposed to military action or exporting the Revolution in certain situations. He certainly saw the value of fostering communist parties in other countries and the invasion of the Baltic republics and Finland for example. But Stalin simply was not concerned with the threat of deterioration of socialism and democracy because that benefited his own grip on power. Better then to seek an accord with the capitalist powers and build his power.

We can also see that as Stalin accrued more power and influence he develops more paranoid and autocratic tendencies. He progresses from demoting his rivals, to expelling them from the party, to imprisoning or exiling them, to executing them, to then executing imagined enemies en masse with his various purges. Instead of seeking to smash the most existential enemy socialists could have in the fascists, he signs a non-aggression pact with them.

Stalin was a man afraid. He rose to power through politicking and scheming, not because he was a particularly inspiring leader or profound thinker. He was afraid of losing control, he was afraid of encouraging other possibly more intelligent or inspiring leaders in other countries, he was afraid of creating war heroes who could challenge him for public approval, and he was afraid of losing a war and being deposed.

Further reading on the development of Marxist thought in Russia in the lead-up to the Revolution, and which details the various schisms and splits and evolution of Bolshevik thinking can be found in Bolshevism: The Road to Revolution by Alan Woods