Usage, meaning and origin of the "swing state".
Solution 1:
The two meanings are not distinct -- to be a "tipping point" a state must have both attributes. In the vast majority of elections a state such as North Dakota has so few electoral votes that it's of little consequence whether it "swings" one way or the other. A state such as Ohio, on the other hand, has a substantial number of electoral votes -- enough to "swing" many elections -- and it also often has a relatively close popular vote.
As to when the term became popular, the earliest valid occurrence that Ngram finds is 1953. Normally one would guess such a term had rattled around for a dozen or so years prior, but a lot of new political terminology came about post-WWII, so likely the term was not more than a few years old then.
Solution 2:
The term "swing state" comes into play only in relatively close elections like those that have taken place for most of the 21st century. It was not an issue in the 1952 election for instance (Eisenhower won by a large margin), but may have been an issue in the 1948 election (Truman won convincingly overall, but won California, Ohio and Illinois by less than one percentage point each). Before that, you need to go back to 1916 to find an election close enough for "swing states," and before that, to the 1890s.
The term "swing state" may go back to the period immediately after the Civil War. That's when party loyalties split along North-South lines. At that time, the 11 former Confederate states (the "Solid" south), tended to vote for Democratic Presidential candidates. To a certain extent, this was also true of the "border" states. On the other hand, the Union states (minus most border states), tended to vote for Republican candidates.
From 1860-1932, the only Democrats (other than Woodrow Wilson who benefited from the 1912 Republican party split), who had a chance to win the Presidency, were the current or former governors of New York. That's because New York was the state with the largest number of electoral votes. The North's demographic advantage notwithstanding, if the "tristates" of New York, Connecticut and New Jersey were removed from the Republican total and added to the southern states as defined in the first paragraph, the Democratic candidate could win. Grover Cleveland did this in 1884 and 1892, losing New York state and the election in 1888. This fact made New York (and the other two) "swing states." More to the point, this phenomenon may have been noticed in 1888, when the successive swings of New York decided the second of two back-to-back elections. Most of the other states were then reliably Democratic or Republican, and were not "swing states."
In 1876, New York's Governor Samuel Tilden won the South plus the New York tri-states, minus Louisiana, South Carolina and Florida, which became "swing states" the other way; and lost the election by one electoral vote, 185-184 to Rutherford B. Hayes. Governor Al Smith could not carry the tri-states in 1928 and won only a handful of southern states. Governor Franklin Delano Roosevelt of New York beat Herbert Hoover in a landslide, but that's another story.
Nowadays the "swing states" have two characteristics: They could go either into the Republican or Democratic column, and they are large enough in terms of electoral votes to make a difference. An example is Virginia, with 14 electoral votes, usually Republican, but went for Obama in 2008 and 2012. Pennsylvania and New Jersey may be considered "swing states" in the other direction. Although "large," Texas is not considered a swing state because it is solidly Republican; New York is considered solidly Democrat.