Best Of

You can find a less confusing definition, @lb_c119, if you click on DICTIONARY in the top left-hand corner of this page.

especially

ADVERB

1 Used to single out one person or thing over all others.
‘he despised them all, especially Sylvester’
‘a new song, written especially for Jonathan’

2 To a great extent; very much.
‘he didn't especially like dancing’
[as submodifier] ‘sleep is especially important in growing children’

From that page you can click to see more example sentences and also synonyms.

But that rather awkward definition that you've found does apply to your sentences.

• The car is quite small — more in the particular circumstance that you have children.
  (This is using small in a relative sense 'small in relation to your needs'.)

• Teenagers are very fashion conscious — true more with girls than with boys.

• I love Rome — more in the particular circumstance when I'm there in the spring.

@lb_c119

Most English verbs produce forms called PARTICIPLES. Generally they are formed with a suffix.

-ING
Almost all English verbs have a form with this suffix.

-ED
The majority of English verbs have a form with this suffix.
An important minority have forms similar in use and meaning but not created with a suffix. For example:
known, gone, taken, written, read

PARTICIPLES may be used like ADJECTIVES in three ways
1. before a noun — a screaming child, a watched pot cf a happy child, a black pot
2. after BE — the child is screaming, the pot is watched cf the child is happy, the pot is black
3.in a phrase after a noun — a child screaming for its mother, a pot watched by the cooks cf a child happy in life, a pot black from long use

In these uses, the -ing forms are ACTIVE and the -ed forms are PASSIVE.
(This important distinction is not reflected in the traditional terms present participle and past participle.)
OK, there are some exceptions, but very few:

• Before a noun, as in an escaped prisoner, a grown boy, the retired manager and the departed guests, the -ed forms are not passive but related to PRESENT PERFECT a prisoner has escaped etc
• After BE some (not all) of these may be used — e.g. the manager is retired

Note that

• before a noun. a participle must generally stand alone as one word
• -ing forms may be used in different ways — as NOUNS or noun-like words
• some -ed forms are not participles but are created from NOUN PHRASES — eg three-legged

For these reasons we have some strange pairs

• a smoking gun (smoking= participle) vs a smoking jacket (smoking = verbal noun)
• a skinned rabbit (skinned = participle) vs a dark-skinned man (from dark skin)

Now many participles have changes to become more like adjectives.

• Some have combined with prefixes — e.g. unsurprising, disinterested although there are no verbs unsurprise, disinterest

• Some have gained additional meanings. One grammar gives the examples
  — She is calculating (but her husband is frank) vs She is calculating our salaries (so don't disturb her while she is doing the arithmetic)
  — They were relieved (to find her at home) vs They were relieved (by the next group of sentries)
  In the first sentence of each pair, the -ed/-ing word can combine with very
  — very calculating (but NOT very calculating our salaries), very relieved (but NOT very relieved by the next sentries)

• Some other -ed/-ing words can be used with very —very very interesting, very willing, very excited, very interested

• Compounds made up another word and an -ing/-ed word can also be used with very — The story is ver heart-breaking, The egg is very hard-boiled .
• Some -ing/-ed words can combine with the suffix -ly to form ADVERBS —willingly, excitedly

I think it's safe to classify any -ing/-ed word that combines with very or -ly as an ADJECTIVE

Now located has not become completely like an an adjective. We can't say very located or locatedly
But there's some difference between The lost city has been located and a small town located 30 miles south of Chicago.
In the first, somebody has searched for and found the city. In the second the small town didn't need to be found; it's where it always was. That difference in meaning must be why the Advanced Learner's Dictionary classifies located as an adjective.

The meaning of be located is 'be (geographically speaking)'.
Both located and the near-synonym situated must be followed by a phrase which is an expression of place. So

• The adjective south may be followed by by a phrase — south of Chicago
• This phrase may be preceded by a phrase expressing distance — 30 miles south of Chicago
• This phrase in turn may follow the word located — located 30 miles south of Chicago
• And finally this phrase can follow a NOUN within a NOUN PHRASE — a small town located 30 miles south of Chicago