One of the construction/architectural dictionaries I consulted says that a pier is one of the pillars supporting an arch. Two dictionaries say that a pier is the support for one of the arches on a bridge. Another reasonably says that a pier is a wood post driven into the ground, below water, to support a structure. The way I've always used the term is described by another dictionary that says that it's the thickened section of a supporting wall which provides lateral support to concentrated vertical loads or, as another dictionary put it, the load bearing brickwork in a wall between openings. Several dictionaries allow that the term is widely used interchangeably with the word column. Or, to describe a large column, or a squat column or a short column.
Similarly the words post and column are used interchangeably. Or, a post is a slender column. Or a post is thicker than a stud. Or a post is a member used as a brace, shore, prop or jack. Or a 4x4 is commonly called a post. Or a post is the load bearing vertical member in post and beam framing
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which divides the structure of a building into bays. One author (and I am grateful to him for his candor) admits that a post is an upright framing member and that the word has many uses.
A column is, of course, a relatively long, slender structural compression member which is used vertically. Of course? Well, no. Of course a column can also be a pier, a post, a pillar, a strut or a stanchion.
What you need to know: You're not wrong. The person you're speaking to is not wrong. There are no nationally agreed upon terms. Actually, I find that rather relaxing.