Is the word "grid" being used in a way I'm not familiar with?
I decided to create an account and try a puzzle, and I selected "4x4 grid" as the difficulty option. I've solved the freaking EINSTEIN logic puzzle, okay, and I was completely flummoxed by what came up. Furthermore, the puzzle was not a 4x4 grid. It was SIX of thema semantical faux pas that I found extremely jarring. The system of interlinked grids also caused me to find the puzzle ridiculously difficult as I had severe trouble keeping mental track of how one logical elimination led to the next. Clearly, I've had no inkling of how logic puzzles truly work at this point, despite solving the Einstein puzzle. Could someone explain just what is going on here to me? Particularly the (what I view as) grossly improper use of the word "grid" in describing the puzzles

The grid refers to the final solution  with a 4x4 grid, your final answer will include something that looks like:
name  trait 1  trait 2  trait 3 name  trait 1  trait 2  trait 3 name  trait 1  trait 2  trait 3 name  trait 1  trait 2  trait 3 In order to solve that most easily, using the given clues, the traits are all crossreferenced; therefore, you get a total of six boxes. It works just like the distance charts found in most atlases. Don't know how to explain it better. This format follows the same standard logic puzzle layout that I've ever seen. Perhaps you are overthinking this, riding the high from that "Einstein puzzle"? :) 
4x4 means:
4 categories (ex. price, drink, food, person) 4 in each category (ex. $4, $5, $6, $7) Only the top row matters, but the other 3 boxes are there to help you figure out what's what by placing the Xs and Os. You can click the clues on the right side to eliminate them when you don't need them anymore, and there's a tab right above where it says "active clues" called "notes" where you can write down things should you need to. But if you think the 4x4 is a monster, check 4x7! (4 categories, 7 options in each category) 
The term "Grid" here is used somewhat loosely. The first number is the number of categories, and the second number is the number of items in each category.
So you are going to end up with some arrangement of squares and each square is defined by the second number. (e.g. in a ?x3 grid you will end up with several 3x3 boxes connected together) The number of boxes is a bit trickier to describe. I hope someone else can come in and give a simpler version than what I'm about to write. You have exactly one box to compare each of the categories with. Let's think about writing the grid from left to right. So we define one category as our first column and then fill in the rest as our rows. Since we don't compare a category to itself we end up with the long side being one less than the number of categories. Then we take the last category in the rows and make it the next column. But we drop of the bottom box where it would be compared to itself. On and on until we end up with only one row being used. If my math is right, the total number of boxes you end up with is (c*(c1))/2 where c is the number of categories. Solving them takes some practice, don't be afraid to use this game's help button. It will give you the next logical move and tell you how you should know it. Ultimately it comes down to knowing that none of the people/objects have anything in common in the categories. So if I know James is wearing a blue shirt I know that no one else is wearing a blue shirt. And since the two are linked, anything I know about the person wearing blue I then know applies to James (and vice versa). 
The "X by Y" format of the grid sizes is as follows:
"X" = number of categories "Y" = number of items to match 
4 x 4 grid
The first number is how many categories there are. For example, the price, the name, the city and the animal.
The second number is how many possibilities there are in each category. Four possible price points, four people, four cities, and four animals. Each combination of categories will produce a 4 x 4 grid, such as: .... Ben . Mia . Sam . Reb $3  ___  ___  ___  ___  $5  ___  ___  ___  ___  $7  ___  ___  ___  ___  $9  ___  ___  ___  ___  The total solving area will have 3 of those grids in the top row, 2 in the second row and 1 in the last row. 
The grid can be seen by clicking on the answers button, next to the clue and notes button. You'll see that it is x columns by y rows.

Cheri from Minneapolis
I find the grids invaluable when doing these puzzles. If you methodically read through the clues marking what you discover clue by clue the grid will point out the answers. Or you could create a grid of your own that you understand. They're really visual tools.

4x4 grid refers to 4 categories each with 4 options per category.There are 6 of these smaller grids to ensure each category can be crossreferenced with another. Hope that makes sense for you :)

The way I understand it, the first number is the sets of clues and the second number is how many of each clue you can choose from. So, a 3x4 grid would be three sets of items (i.e. names, dates, and mementos) and four of each to choose from (e.g. Nancy, Tom, Earl, and Tina for names). So, yes, it is kind of confusing at first, as you might expect just three lines across and four lines down, but it's just the way it seems to be. Hope that helps.

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