If you place the cursor within the centred table and open the Table Designer, does it actually have the Left setting?
A screenshot (wih Broders & Text Symbols enabled) would help.
> ... specific style ...
Style of what? Paragraph?
If so, the alignment of the anchoring paragraph is independent of the alignment of the anchored table itself (in Table Designer).
This may or may not relate to your case...
Does your text flow allow for side heads? If so, then if your table anchor is in a paragraph that is in the text column (that is, NOT in the side head), then the table will also be in the text column IF it is narrower than the text column. Such a table aligned left aligns at the left side of the TEXT column, NOT the left side of the text FRAME. If you make the table wider than the text column, it will align to the left side of the text frame.
If you want a narrow table to align with the left side of the text frame, then put its anchoring paragraph in the side head.