Generally the former.

In general, up until security began tightening significantly after McKinley's assassination, you'd show up, and if the gate guard was present and didn't turn you away (which by the late 19th century they did quite a bit more often), you'd present your card to the President's private secretary. Then, if the President felt like it, you might have a few minutes with him.

I don't think I've ever run across the protocol for Philadelphia - although Adams spent a good deal of his time in office back in Massachusetts given the (not entirely unreasonable) fear of yellow fever - but once the government moved to Washington in June of 1800, given how primitive the conditions were there during his administration, he'd probably have been happy to receive you. You'd have been one of the few people blessedly not part of the government, and he'd probably been eager to sit down with someone new if only to learn the most recent news from wherever you came without it being filtered through a very partisan press.

Save for wartime, up until the 1900s security at the White House most often consisted of a few guards - often from the Washington police, sometimes privately hired - and a doorkeeper and private secretary. Even that came under challenge from the Adamses - the younger of which discussed seeing routine random visitors between 11-4 - and Jackson. (Monroe was one of the few early presidents to favor it and constructed the first fence around the White House as well as restricting visitors from walking around it without escort.) Even this small amount of protection didn't become a permanent arrangement until Tyler.

Also, one thing to keep in mind is that until the 1840s when daguerreotypes became routinely printed in the press, most people outside of the immediate circle of government didn't really have much of an idea of what the President actually looked like - hence why you'll read stories about some of the early ones wandering around town without any escort at all. For instance, there's a story about Jefferson running into a vitriolic critic of "The President" on the streets of Washington, listening to him rant a while, and then rather surprising him with an invitation to the White House.

We don't have much on how Adams in particular handled visitors, but once they were called in they were generally seen without bodyguards and often alone; remarkably, during the first few weeks of his term following Lincoln's assassination, Andrew Johnson received callers at his temporary lodging (he was waiting for Mary Todd Lincoln to move out of the White House) without so much as a butler, let alone guards, during the meetings despite being a potential assassination target himself. Going back to Jackson, his protocol is interesting because it's documented and seems to reflect what generally went on at the White House until after McKinley, where by the 1880s something like 100 people a day visited even if most didn't see the President. From Ayton's Plotting to Kill the President:

"Visitors to the White House who wished to see Jackson would go to the porter’s lodge and sign the visitor’s register. The steward would take the visitor’s name and give it to a servant, who would take it to the president or his private secretary and nephew, Andrew Donelson. If the person was approved, he or she would be taken up the stairs and announced. If the visitor was well known to Jackson, that person might be taken immediately to see him. Some visitors arrived at the White House without an appointment. The doorkeeper would usually take the visitor’s card upstairs to the president. Visits by ordinary citizens were usually brief, but sometimes took up a whole morning of the president’s time—office seekers, in particular."

Jackson is of course (in)famous for his unintended bacchanalian inauguration reception, but in general there was a protocol established for the "public" receptions. They'd be advertised, but also widely understood that only "certain elements of society" would be welcome; the common riffraff just stood at the gates and gawked, and occasionally they'd throw mud at the carriages. The White House lawn does seem to not have had the same restrictions until Hayes (who also inaugurated the Easter Egg roll), which went back and forth a bit; Cleveland closed the south grounds, McKinley opened it back up.

All this of course changed with Teddy Roosevelt, even if TR delighted in trying to leave the Secret Service detail forced upon him behind; the fear of anarchists was real. Even then, unlimited access to the White House was slowed significantly but didn't stop; like most of Washington of that era, he would receive cards from callers doing their social rounds - in his case, between 9-10 pm. This availability was slightly reduced in Wilson's first term - he wanted to continue the open door policy but had a large number of "cranks" (the term that had been used since the mid 1800s for mentally ill individuals obsessed with the President, although unlike his predecessors, many of Wilson's were oddly enough women) try to see him - but the modern era really began with the United States' entry into World War I closing the White House to the public. Harding eventually opened it back up to visitors in his return to normalcy, but that was more in the realm of self guided tours than a random citizen getting in to see the President.

Interestingly, one of the reasons Harding went into politics was his father, who took advantage of the previous policy while serving in the Union Army when he was transiting Washington. He simply walked in to a White House that was guarded externally far more than in peacetime, talked to one of the secretaries who warned him he probably would wait for hours with no guarantee, but after 6 or 7 hours he and another member of his company got to meet Lincoln for 5 minutes. It was a highlight of his life.