A ‘wedge’ as understood by interest rate options traders is a structure of the form : long a cap/floor straddle struck ATM for a period of 1 yr starting in N years / short a N year into 1 year swaption straddle also struck ATM. Usually the cap/floor underlying is 3mo Libor but nowadays it can be daily SOFR.
This transaction has two principal exposures (a) it is short the correlation between the short term rates (either quarterly Libor or daily SOFR ) and (b) it is long volatility , specifically the forward vols of the short term rates that remain after swaption has expired.
‘Wedge’ is indeed a term recognized by the trading community. If you draw a diagram of forward vol vs calendar vol , the region of exposure of this trade is represented by a triangle that looks like a ‘wedge’.
To answer the comment of @JUW: yes this is well expressed in HJM framework. In that model, correlations are defined as between pairs of short term rates. Therefore as you say , (a) comes only from the swaption. However (b) is the net exposure (eg cap/floor is long 120 units of Vega/ swaption is short 100 units).