î in the middle of a word
so if you should use â in the middle how come there are many words (names) that have î in the middle (for example Țîru, Hîncesti, Rîșcani)
This rule dates back to only 1993.
Before that, everything was written with "î", except for the name of the country, nationality, and language (and all derived words).
While this reform was pretty drastic, you can imagine that it's quite difficult to change proper names, like going to a person and telling them they should change how they're called. So a lot of names still follow older orthographical conventions.
Long story short: out of fanaticism for Latin, to keep more of an etymological spelling.
It's worth bitterly noting that it's not even that good since there are some words which should be spelled with the other letter:
înger - from latin angelus
râu - from latin riverus
Besides the proper names that retain the original spelling, there are also many compound words that are formed by adding a prefix to a word starting with î. They will retain the initial î even though it is not the first letter anymore.
încredere (trust); NE + încredere = neîncredere (distrust)
încărcat (loaded); RE + încărcat = reîncărcat (reloaded); NE + încărcat = neîncărcat (not loaded); PRE + încărcat = preîncărcat (previously loaded)
a întâlni (to meet); a RE + întâlni = a reîntâlni (to meet again)
Please note that in the last example there are present both î (first initial letter) and â (in the middle) in the base word and they remain unchanged in the compound word.