You were AT the court because you had been summonsed, and you pleaded guilty, then were fined and charged costs. The lawyers did not have to put any evidence before the judge, because your guilty plea said, 'Yes, I did it.' You were convicted of this, and it is on your police record.
All this may not be what you wanted to have happen, or what you understood had happened, but it HAS happened now, and can't be changed.
To deal with the situation for INZ, you will need to have to accept that the incident was careless driving. It will go against you if you keep saying that it was because of the conditions, and not your fault - the road rules make the driver responsible for controlling the car in a way that suits the conditions, and you didn't do that. It is CALLED "careless driving" in law, even though you didn't feel you were being careless.
You could explain to INZ that you did not understand that what happened to you at court was a conviction. Maybe this is because legal procedures are different in your own country, so you didn't realize what was happening. I expect this is how come you did not fill in the ITA correctly - you didn't know you had a conviction. I think it's important to apologize for the mistake on the form, and assure them that you did not have any intention to mislead them - as they can see, you mentioned the incident on the ITA, just did not describe it correctly.