Max-Width vs. Min-Width

In short, min-width is a mobile 1st approach, max-width is a desktop 1st approach.

Min-width is the minimum width at which a style will START to be applied. (Have to be ordered from smallest to largest to work properly, regular styles first). Put another way: If device width is greater than or equal to..., then apply some specific styles. With min-width, styles START and continue forever as long as min-width is met, and no max-width is specified.

Max-width is the maximum width at which a style will continue to be applied. After that, the style will STOP being applied. (Have to be ordered from largest to smallest to work properly, regular styles first). Put another way: If device width is less than or equal to..., then apply specific styles. Styles STOP as soon as width greater than max-width is hit.

Finally, It depends on how you want to implement. There is no ONE RIGHT solution as some may claim. In my opinion min-width works great when starting from scratch, but max-width often makes more sense to me when retrofitting an existing web site.

2 Part Answer

Part 1: To answer "why people are using min-width over max-width?":

It has to do with design flow. Typically, with min-width patterns, you're designing mobile-first. With max-width patterns, you're design desktop-first.

Going mobile-first with min-width, the default style is the mobile style. Queries after that then target progressively larger screens.

body {
    /* default styles here, 
       targets mobile first */
}

@media screen and (min-width:480px) {
    /* style changes when the screen gets larger */
}

@media screen and (min-width:800px) {
        /* And even larger */
    }

Conversely, using max-width, is desktop-first then adds queries to make styles mobile-friendly

body {
        /* default styles here, 
           targets desktops first */
    }

@media screen and (max-width:800px) {
    /* style changes when the screen gets smaller */
}

@media screen and (max-width:480px) {
        /* And even smaller */
    }

Part 2: For your particular custom navigation for any device who's width is 360px or less:

You could include that as a separate max-width query, IF thats the only exception to the rule. OR use that style as your baseline, then change it for wider screens.

If you do an exception (which isn't really following mobile-first design methods), it'd be something like:

body {
        /* default styles here, 
           targets mobile first
           ALSO will cover 361 - 479 width */
    }

@media screen and (max-width:360px) {
    /* style ONLY for screens with 360px or less width */
}

@media screen and (min-width:480px) {
    /* style changes when the screen gets larger */
}
etc...

It really depends on how your stylesheet works. For example:

@media screen and (min-width:100px) {
    body { font-weight:bold; }
}

@media screen and (min-width:200px) {
    body { color:#555; }
}

The above two media queries would make the body font bold if the screen is greater than or equal to 100px, but also make the color #555 if it's greater than or equal to 200px;

Another example:

@media screen and (max-width:100px) {
    body { font-weight:bold; }
}

@media screen and (max-width:200px) {
    body { color:#555; }
}

Unlike the first example, this makes the body font bold and color #555 only if the screen width is between 0 and 100px. If it's between 0px and 200px it will be color #555.

The beauty of media queries is that you can combine these statements:

@media screen and (min-width:100px) and (max-width:200px) {
    body { font-weight:bold; color:#555; }
}

In this example you are only targeting devices with a width between 100px and 200px - nothing more, nothing less.

In short, if you want your styles to leak out of media queries you'd use either min-width or max-width, but if you're wanting to affect a very specific criteria you can just combine the two.