No, the trigger button will not "focus as well". Since you are already holding the AF-ON pressed, the camera's autofocus is active. It can't do any more than that. It can't be active-active. Thus you'll not notice any difference.
But it looks like you have set AF-ON to also switch to Servo AF when held. And if so, then that setting does have priority, so as long as you keep AF-ON pressed, then Servo AF it is.
However, if you release the AF-ON button and have focus active on the trigger button, then you get all that others have already commented about.
Now this is how I was thinking it was. Cyberdyne is saying the opisite as is the person who got me thinking about it and lead me to my post.