Edgy style in fashion means having the cutting edge style, being trendy or voguish, and leading the latest fashion.
Avant garde in fashion means at the fore-front of the fashion trend setting.
Edgy style and Avant garde in fashion have similar meaning, being at the fore-front, being innovative, and leading the fashion trend.
Slightly punk means a little rebellious to current trend, it is innovative for being different, but is not necessarily trend setting, which will mean gaining the acceptance of the general public over time. Therefore, edgy style does not have the same meaning as slightly punk.