There are two varieties of Khiriq: Short and Long. A long Khiriq ends a syllable, like in the word רִאשׁוֹן (Ree-shon). A short Khiriq comes in the middle of a syllable between two consonants, like in the word יִצְחָק (Yeeṣˤ-khaq).
According to the rules of Lashon that we use today, a khiriq is normally short. If it’s followed by a ga’ya or by a silent letter, it’s long.
Thus, the words רִאשון, כִי, and פִיךָ all have a long-Khiriq, while מַיִם, מִן, and בִמקוֹם have a short-Khiriq.
Many words have a long-Khiriq followed by a consonant. In such words, the consonant is pronounced as if it were a separate syllable. For example, פנִים is pronounced /Pa-nee-m/.
In many communities—both Mizrahi and Ashkenazi—a short-Khiriq is pronounced like the English short-I (like in your example “pit”) instead of “EE”. This is an entirely valid Minhag, but it’s not the original way.
You brought נִקּוּד in your question. In this word, the letter after the Khiriq—ק—has a Dagesh hazzaq. This means that the letter is doubled, as if it were two, by being pronounced slightly more forcefully. The word is thus broken up as /niq-qud/, with the lengthened ק being part of two separate syllables.
This Khiriq does not have a ga’ya, and does not end the syllable, as shown. Thus, it is a short-Khiriq.
Note: All the above is based upon the rules that have been in use since the Medieval Era. Before then, the rules were slightly different.