How to post images on my blog with a pre-defined border

this is more a css problem than wordpress related –

check style.css of your theme: the ‘border’ (actually a padding with background image) is only set for left, right, and center aligned images (like those from your first link), not for those with .alignnone (those from your second link):

#content img.alignleft, 
#content img.alignright, 
#content img.aligncenter {
    background:url("images/wf-hashes-wht.gif") repeat scroll center center transparent;
    margin-bottom:12px;
    padding:6px;
}

if you want the same behaviour for non-aligned images, add #content img.alignnone to the above list (don’t forget to add the comma)