The phonetic position of vowels in English really depends heavily on the accent. Note that the chart is from a 1982 source, so even ignoring that different speakers use different qualities for vowels, there's been plenty of time for changes to have occurred. You should just think of the chart as showing approximate areas where you'd find vowels. I quote two other charts showing the different positions of vowels in different accents of American English in my answer to /ɑ/ vs /ʌ/ pronunciation (those are from a more recent, 2005 source).
The use of the phonemic symbol /ɑ/ to represent a phoneme that is often realized as a phonetically central vowel could be explained in various of ways: convenience or simplicity of symbols used in transcription is one, consistency with the transcription of other dialects of English (past or present) where /ɑ/ is phonetically a back vowel is another. If there are any theoretical reasons for considering this to behave phonologically as a "back" rather than as a front or central vowel phoneme in the English vowel system, that would be another explanation (I can't think of much that would support that analysis, but there might be evidence for it).
From a notational standpoint, it's considered preferable to distinguish phonemes in transcriptions with different IPA symbols rather than just with diacritics.
The chart is attributed to John Wells's Accents of English, and something else that you need to understand to interpret it properly is that this source makes heavy use of phonemic transcription in order to make it clear which sets of words are being talked about. It would be redundant on a phonetic chart to use the strictly phonetic symbol for the point that you are marking; using the conventional symbol "ɑ" is a way of unambiguously marking that the vowel that is shown at the bottom of the chart is the vowel used in words like lot, stop, sock, father, bra, spa (examples taken from pp xviii-xix, where Wells explains the book's "Typographical conventions and phonetic symbols").