Confused non-ref here. I have heard announcers talking about breaking the plane of the goal line. Watching games it appears that the goal line plane is on the field side of the white line.
When the ball is placed on the field after a touchback the nose of the ball appears to be placed touching the edge of the white line. That also happens going the other way.
1. Where is the plane of the goal line?
2 Where is the plane of a yard line in the field of play? The middle of the line, or at the leading edge depending on which direction the team is moving down the field?
You are correct that the "planes" of the goal lines are the field side edges of the goal lines. They are exactly 100 yards apart (in theory, anyway

). Any time a live ball in player possession of an 'attacking' player touches or penetrates (in part or in whole) this plane, a touchdown is scored. Conversely, if the ball is dead by rule in the possession of the defending team, and the ball is touching or penetrating (in part or in whole) this plane, that is either a safety or a touchback (dependent on impetus).
The yard lines are just lines - not planes. All yard lines are 4" wide, and centered on the lines that are at 5-yard intervals between the goal lines. You have discovered a slight idiosyncrasy with the layout of the field, in that, for example, the edges of the 20 yard lines nearest the goal lines, are actually only 19 yards, 2 feet, and 10" from the goal lines.
Following a touchback,
by rule, the ball is placed at the 20 (or 25)
yard line - not 20 (or 25) yards from the goal line. That's why you see the nose of the ball right on the edge of those lines following a touchback. Note that thereafter, when the ball is advanced on or beyond any part of 30 (or 35) yard line, there is no need for a measurement - that's a first down.
FYI, you may also notice that, following a change of possession, like a kickoff, punt, interception, fumble, etc., the game officials will try to, similarly, place the ball on a yard line, and not leave it between yard lines. That is mostly for convenience in establishing the line-to-gain exactly on the yard line 10 yards away. Again, if the ball is advanced on/beyond the line to gain, no need for a measurement. Coaches know this, and are OK with it, because it helps maintain the "pace of play." (Anybody heard that term recently?)
Hope this helps.
Robert