Wrong! Or, at least, in Lincoln's case, you're both kind of correct. Lincoln had, at the insistence of Secretary of War Edwin Stanton, at least one company of Union cavalry assigned to his protection. This was in addition to Ward Lamon, a personal friend of Lincoln's from his lawyer days who took it upon himself to act as his bodyguard. Indeed, Lamon and Pinkerton clashed over the President's security even before the war broke out. Pinkerton claimed an assassination plot was afoot in the lead up to Lincoln's inauguration in early 1861. There is some debate over the validity of these claims, as Pinkerton's intelligence gathering capabilities seem to have been greatly exaggerated. Regardless, the cabinet took these claims seriously enough to change Lincoln's travel plans at the last minute, and he arrived in DC under the cover of night and with only Lamon present with him during the legs of the journey where Pinkerton claimed the most danger lie. Once inaugurated, and in the midst of the war, his protection became a military matter, which is when the cavalry companies stepped in. However, he routinely delighted in giving them the slip. In 1864, someone took a pot shot at him whilst riding through DC, and he was forced to take his protection more seriously. It was at this point that the DC police assembled a 4 man permanent bodyguard detail.
So, depending on when exactly you're looking, Lincoln's protection detail could have consisted of 2 companies of Union Army troops, Pinkertons, a self-appointed bodyguard cum lawyer pal, 4 police officers, a Freedman valet/bodyguard (William Henry Johnson), and a partridge in a pear tree.