Words like city and county often appear in proper nouns alongside their use as common nouns. Most style guides recommend capitalizing these words only when used in full proper names. For example:
Wayne County is a county in the U.S. state of Michigan. As of the 2010 census, its population was 1,820,584 making it the 18th most-populous county in the United States. The county seat is Detroit, the largest city in Michigan.
Note how the entry only capitalizes Wayne County.
According to the 2000 census, the county has a total area of 672.20 square miles....
This last case is what the Illinois Identity Standards mean by “shortened forms of official names.” Even though it specifically refers to Wayne County and might be considered an abbreviation of the proper noun, you do not capitalize county when using it alone like this. Likewise, you would not capitalize the city or the county seat even when referring specifically to the city of Detroit.
You can read many such phrases in more than one way. City of London is a famous example: The City of London is a district of the city of London. Capitalization always follows from the phrase's meaning, not its form. It would be incorrect to capitalize city of London when referring to the greater metropolitan area. Likewise, the Research Triangle is anchored by “the cities of Raleigh, Durham, and Chapel Hill” even though City of Raleigh appears as a proper noun in official documents.
Often, a common noun is more appropriate even when it exactly matches the form of a proper name. Follow the meaning, not the form. For example:
The Commonwealth of Virginia has many natural reservoirs. The Ragged Mountain reservoir is located closest to the City of Charlottesville.
Here, City fits the meaning; even better, it matches the formal register of Commonwealth.
The Ragged Mountain reservoir is located closest to the city of Charlottesville (but the unincorporated community of Ednam is closer).
In contrast, this example emphasizes city as a geographical role and not as part of Charlotte's identity. Therefore, it requires the uncapitalized common noun.
You capitalize it when it is used as part of a proper noun. Your first, second, and fourth examples are correct, the third is not.
He gave two examples of large lakes, Michigan and Superior.
In this example, the Lake part in the proper nouns Lake Michigan and Lake Superior is omitted. The word lakes here is not part of a proper noun. So capitalizing it like this,
He gave two examples of large Lakes, Michigan and Superior.
Is incorrect.
In other cases, the same rule applies: capitalize the words when they appear as part of a proper noun, such as Mount McKinley or Round Table, but not when they appear as common noun, such as the tallest mount/mountain or coffee table.
It is also recommended to capitalize references to specific figures, tables, chapters, sections, equations—particularly when they appear with a number or letter identifying them:
This trend is demonstrated in Figure 5 in Appendix C
But again, not when simply using the common noun.
This trend is demonstrated in figure below.
Best Answer
The Guardian style guide says:
So:
Becomes:
Which is also a lot easier on the eye.