peinlich
embarrassing
Causing a feeling of embarrassment, shame, or discomfort.
Es war mir furchtbar peinlich, als mein Handy im Theater klingelte.
I was terribly embarrassed when my phone rang in the theater.

embarrassing / meticulous
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embarrassing
Causing a feeling of embarrassment, shame, or discomfort.
Es war mir furchtbar peinlich, als mein Handy im Theater klingelte.
I was terribly embarrassed when my phone rang in the theater.

The word 'peinlich' (embarrassing / meticulous) is derived from 'die Pein' (torment, pain). This explains the old meaning related to torture and also the modern meaning: an embarrassing situation causes a kind of emotional pain.
Sein unhöfliches Verhalten war eine Pein für die ganze Familie.
His rude behavior was a torment for the whole family.
To express for whom a situation is embarrassing, you use the structure 'Es ist + [dative pronoun] + peinlich' (It is embarrassing for [someone]). The subject of the sentence is often 'es' (it).
Es war ihm sehr peinlich, dass er den Namen vergessen hatte.
He was very embarrassed that he had forgotten the name.
In the meaning 'übertrieben sorgfältig' (overly careful), 'peinlich' is almost always used as an intensifier before another adjective such as 'genau' (precise) or 'sauber' (clean). The most common combination is 'peinlich genau' (meticulously precise).
Sie achtet peinlich genau auf die Einhaltung der Regeln.
She pays meticulous attention to compliance with the rules.
When 'peinlich' stands alone, it almost always means 'unangenehm' (embarrassing / unpleasant). When it stands directly before an adjective such as 'genau' (precise) or 'sauber' (clean), it means 'extrem' (extremely) or 'übertrieben' (excessively).
Die Stille im Raum war peinlich.
The silence in the room was embarrassing.
The meaning 'connected with corporal punishment' is outdated. Today you will find it only in historical texts, often in connection with the legal system of the Middle Ages.
Die „peinliche Gerichtsordnung“ regelte schwere Strafverfahren.
The 'peinliche Gerichtsordnung' (penal court ordinance) regulated serious criminal proceedings.
exact / precise
'genau' and 'peinlich' can both express that something is done very carefully and precisely. 'genau' is the normal word for exact, correct, or without errors. 'peinlich' has this meaning almost only in phrases like 'peinlich genau'. Then it sounds stronger and means: someone pays attention to every small detail. The words can be confused because 'peinlich genau' means something similar to 'sehr genau' (very precise). By itself, however, 'peinlich' usually means embarrassing or unpleasant, not precise.

Der Prüfer kontrolliert die Liste peinlich genau vor dem Start.
The inspector checks the list meticulously before the start.
Bitte lies die Adresse genau, damit wir nicht zum falschen Haus fahren.
Please read the address carefully so we don't drive to the wrong house.