Marking studs with tape measure
When you start a wall, your first board is not supposed to be on 16" centers. You're right, the first stud will end up  15.25" center to center from the second stud but that is because you don't want you first stud to be centered on the edge of the sheeting (whether your sheeting it with plywood, sheetrock, or whatever you're using). Rather, you want your sheeting to be flush with the edge of your first stud. So yes, when you measure from the beginning of a wall, mark it the way that you've been reading.

Your thinking is correct when you're in the middle of a wall. If you hook the tape measure on the edge of one stud, your 16" line should come up to the edge of your next stud. Studs that are on 16" centers are 16" from the center of the one stud to the center of the next stud. So if you are building a long wall, and you are not measuring from the beginning of the wall, you would hook your tape on the board where a stud is and put your each multiple of 16. Put your "X" on the far side of the line and bury the "X" when you put your stud in.

Just remember - for every wall, there is one exception... the first stud.