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The formatting of the original/imported dates and the hand-typed dates appears the same (X/X/XXXX), except the hand-typed dates are displaying as aligned right, and the imported dates are displaying as aligned left. Both are set for General horizontal alignment.
That's the key: the imported dates are probably coming in as text, not numeric. It does not matter what the numeric format is; you can still enter text.
Why are the imported dates coming in as text? My guess: they include spaces or non-breaking spaces. The latter are common when you copy-and-paste from web pages.
If they only contain spaces, you can remedy the problem easily by using the Text To Columns feature.
If they contain non-breaking spaces, you need to use SUBSTITUTE to rectify the problem. The following corrects both problems:
=--SUBSTITUTE(TRIM(A1),CHAR(160),"")
formatted as Date. You can use copy-and-paste-special-value to replace the original imported dates with the SUBSTITUTE result.