I don't think Jackson actually got his hands on would-be assassin Richard Lawrence, but he definitely tried. Nonetheless, it was a legitimate example of someone trying to assassinate a president. Why would it not count?

Although Lawrence was later deemed insane, speculations of conspiracy immediately circulated through Washington. Jackson himself was convinced that Mississippi Senator George Poindexter had arranged the attack. Poindexter immediately proclaimed his innocence and even demanded a Senate panel investigate the matter. Ultimately the Senate did check into it and concluded that Poindexter had not been involved in any plot.

While Lawrence's attack against Jackson was almost surely a random event, it proved as early as the 1830s that presidents---like all authority figures---can be the targets of violence and that Americans in the 19th century were very much aware and concerned of conspiracies and plots to harm public figures.

The possibility of violence against a public figure was not novel in 1861.