This is more or less true, but a touch more nuanced than your summary.

You are correct that it was commonly seen as undignified and un-Presidential to campaign for yourself. Typically, you would be nominated for the candidacy by your party at their national convention - national conventions were first held by the Federalist Party for the 1808 and 1812 elections, then commonly held after that - and then, once you were nominated; you would largely keep your head down and stay out of the newspapers while your friends and fellow party members “stumped” on your behalf. Prominent businessmen and lawyers would travel the areas in which they were well-known, giving speeches from tree stumps (hence the term) about you, often singing campaign songs and otherwise drumming up support.

This process was extremely practical in the days when the population was spread out in small clusters, especially before the advent of better transportation. A Presidential candidate could not hope to cover as much ground as dozens of stumpers, especially when some of these trips would take several days and reach a crowd that otherwise would not have been worth the effort. This had the added political benefit of drawing attention to a candidate while ensuring that mistakes or follies could be blamed on the stumper. If your friend gives a rousing speech explaining why you should be the President, that reflects well on you; if that same friend gets fall-down drunk in a tavern and punches the bartender, your campaign manager can quell the story and distance him from you.

Of all people, little-known President William Henry Harrison was the first to actively campaign for himself in 1840. Harrison’s opponent, then-President Martin Van Buren, and his Democrat supporters, presented Harrison as an old drunk, claiming “Give him a barrel of hard cider, and ... a pension of two thousand [dollars] a year ... and ... he will sit the remainder of his days in his log cabin”. Harrison and the Whigs embraced the image and painted the candidate as a down-home, folksy man running for the common people; essentially, the first President you’d like to have a beer with, the “log cabin and hard cider candidate”.

Ironically, Harrison had been raised in wealth and Van Buren in poverty, but the two men shifted economic positions through their work later in life, but that’s neither here nor there. Part of Harrison’s embrace of the grass-roots image meant traveling and speaking to the common people on his own behalf, since he had more energy and was more connected to the people than that privileged stuffed shirt in the Presidential Mansion. The massive economic downturn of Van Buren’s Presidency did not help his cause, and he was defeated in 1840. Despite Harrison’s short Presidency, he nevertheless left a mark on American politics and all future Presidential candidates would follow his example to one degree or another. The precedent was set, that anyone running for President had to make at least some effort to speak to the common folk, and as transportation improved and more people flocked to cities as a result of the Industrial Revolution, it became much more practical for candidates to speak in their own campaigns, and as a result, more commonplace and less stigmatized.

As to your question about the desire to be President, this was generally more of a case-by-case basis. Following the example of Washington, naturally, it was likewise seen as undignified for the President to want to be President, since that was callous and uncouth, but the truly great Americans would shoulder the burdens of their countrymen in service. This was...mixed, to say the least. For some men, such as Andrew Jackson, it was quite true. Jackson was generally opposed to the idea of becoming President, but nonetheless ran and carried out his two terms faithfully. For other men, such as James K. Polk, it was a facade. Polk sincerely wanted to be President and approached it with gusto, but kept up the appearance of humility for the populace.

In the early days of the United States, many people were still terrified that the Presidency was only a monarch with a different name, and too closely remembered their own experiences or stories from their grandparents about life under British rule, so they were leery about anyone who openly sought power. Over time, as the country moved farther away from the American Revolution and British rule faded from public memory, people became less wary of someone desiring power for themselves. Much of this was helped by the increasing clarification of what happened when, say, a President died in office or was impeached, and the limitations of the office became more apparent. It’s difficult to pin down an exact time and place when it became socially acceptable to say you wanted to be President and no one would see that as a red flag, since it was much more of a gradual process over many decades, but by the early 1900s, it was all but guaranteed that anyone running for President actively desired it.

I’m at work right now and don’t have access to any of my books, so my primary sources are a bit non-existent right now, but I can update this evening.