Tower Controller reported a commercial aircraft was near touchdown when a go-around was announced by the pilot. The Controller expected the air carrier to preform the published missed. However the aircraft was on a visual approach and executed a visual missed approach. The Controller states the airport had just changed to the published missed with the TRACON; due to the weather; and the air carrier was not informed. The Controller had suggestions to remedy this particular situation.

Narrative

I was the Controller providing service. Aircraft X; was attempting to land Runway XXR. We had high cross winds (gusting 25 kts.) today with multiple go-arounds before this one. Aircraft X was over the Runway very low and encountered a gain of 15 kts. They went around on their own and my instruction to them was 'execute published missed'. I did the standard go-around coordination with TRACON then gave the pilot a frequency change. I noticed out the window how quickly the pilot turned for ZZZ1 and commented to the Tower Cab.The problem with the event was Aircraft X was on a visual approach when I thought he was on an ILS approach. We just changed our procedure with TRACON from issuing ALL go-arounds 'Runway heading at 10000' to issuing on visual approach 'Runway heading at 10000' and ILS approach 'Execute missed approach'. When I instructed the Pilot to 'Execute missed approach' he confidently said 'Roger' which further convinced me he was on an ILS approach because he spoke like he was familiar with the go-around procedure he was about to complete.Before we changed this LOA with TRACON I had zero go-around errors for over 4 years at ZZZ. I personally like the simplicity of giving all go-arounds 'Runway heading at 10000' and do not see the advantage of this change. I observed the Pilot overcome a very critical phase of flight and begin their go-around and did not want to add to their work load by verifying which approach procedure they were on. I understand this should have been verified while the plane was on approach but in the event that Controller misses it; or the data tag is inaccurate; it shouldn't immediately lead to a loss of MVA separation.I recommend changing the LOA with TRACON back to issuing all IFR go-arounds 'Runway heading at 10000.'