Ok I understand the VP part, but how does that relate to being part of the house of representatives? That's got nothing to do with a presidential term?
Okay, after re-reading the information, the "unexpired" period relates to whatever remains of a deceased or removed president's term, with the VP assuming the presidency and completing the remainder of this unexpired term.
The link I posted earlier on the 22nd Amendment explains it thus, using Lyndon B Johnson and Gerarld Ford as comparisons:
Johnson became president in November 1963 upon the assassination of John F. Kennedy, served out the remaining 1 year and 59 days of Kennedy's term, and was elected to a full four-year term in 1964. Four years later, he briefly ran for a second full term, but withdrew from the race during the party primaries. Had Johnson served a second full term—through January 20, 1973—his presidency would have lasted 9 years and 59 days; as it happened, Johnson died two days after that date.
Ford, who became president in August 1974 upon the resignation of Richard Nixon, served the remaining 2 years and 164 days of Nixon's term, and ran for a full four-year term in 1976, but lost to Jimmy Carter.
Johnson was eligible to be elected to two full terms in his own right, as he had served less than two years of Kennedy's unexpired term, whereas Ford was eligible to be elected to only one full term, as he had served more than two years of Nixon's.
There's some discussion on the link on interactions between the 12th and 22nd Amendments, ie, can a former two term president subsequently serve as VP, with the potential of assuming the presidency? - but nothing that I can see on a former one term president seeking an elected office below the presidency.
However, a quick google search reveals a precedent: Quincy Adams served as a senator after being a one term president.
This response on quora seems to suggest that Trump could indeed run for either Congress or the Senate; but that most former presidents, if given the option, would consider Congress as too much of a step down. It also references the Presidential Succession Act, if anybody has a mind to check into it further.
The bottom line is, most presidents fulfill two terms; and even one term presidents likely have enough to keep them busy, so as not to seek public office again after being rejected by the electorate. Neither of these really apply to Trump, as he's clearly got plenty of time on his hands, and he doesn't consider himself roundly rejected by the American people.
I hope this clears things up a bit!
EDIT: An extract from Quincy Adams' wikipedia page:
Adams grew bored of his retirement and still felt that his career was unfinished, so he ran for and won a seat in the United States House of Representatives in the 1830 elections. His election went against the generally held opinion, shared by his own wife and youngest son, that former presidents should not run for public office. Nonetheless, he would win election to nine terms, serving from 1831 until his death in 1848.[
Looks like he served in Congress rather than the Senate, but it's cleary there's nothing set in stone on this - simply a precedent or unwritten rule. And we all know how Trump loves to trample this stuff.
