Unfortunately, no. The X button select isn't impacted by any filters you can set. It does have some built in preferences (for instance, although it can select players, it won't if there's an item in range, preferring the item instead, which is useful if you want to select a summoning bell that has a crowd of people surrounding it). But you can't configure it to make its selections based on what you're trying to do. (I keep wishing I could make it ignore mobs when I'm on a gathering class and trying to select the next harvesting node.)
Under Character Settings -> Hotbars, you can select which cross-hotbars are included in that cycle. If you only include one of them, then it won't have anything else to cycle to. (Be sure you're checking / unchecking the boxes for cross-hotbars and not the regular hotbars.)
The cycle can also be different depending on whether your weapon is drawn or sheathed. I use that to include skills like Teleport, Return, calling a mount, and so on, on a bar that won't be used while I'm fighting, because that bar isn't on my "weapon drawn" cycle.
Once you acquire more combat skills than will fit on one bar, you'll want access to more, ideally without having to cycle between them. For access to other bars besides the one(s) that R1 cycles, you can set one to be an expanded crossbar accessible by pressing R2+L2 or L2+R2 (the order matters for which side of the bar comes up) and can set one as W crossbar accessible with a double tap of either R2 or L2.
You can also use R1 + any of the eight face buttons to bring up a particular crossbar as your main bar, regardless of whether it's one you'd normally cycle to or not. (When arranging where skills should go on your bars, you'd use this to bring up a bar normally used as extended or W bar into your main crossbar position, so you can assign skills onto it.)
Under HUD Layout, choose one (or more) of the bars, and there's a show/hide toggle (I think it's the triangle button, but I'm not in game so can't double check that right now. Anyway, it shows a button guide that will tell you which button controls the show/hide toggle.)
HUD Layout is also where you'd position the bar where you want it to go. There's another setting, though, that's in Character Configuration instead. Under the Char Config window's Hotbars section you can select whether you want the hotbar to show up as a horizontal row, a vertical row, a block of 3 by 4 buttons, or whatever.
For controller players, it's mainly used for display purposes, so you can see when skills come off cooldown if they're on a different cross-hotbar than the one you're currently showing. They can be used directly with a keyboard or mouse click.
By default, PS4 uses a feature called "Direct Chat", designed for players (like me) who only use the keyboard for chat and use controller for everything else. With that enabled, whenever you start typing, it will automatically move focus to the chat box and enter what you type there. By disabling it (It's in the General section of Character Configuration), you can use keyboard keybinds for hotbar skills, opening windows, and so on, but then you have to manually move focus to the chat tab before you can type messages there.