A couple of things people should learn:
1) The only claim to territory or political power in any dictatorship is that of the ganglord. Any other group with great control of force can remove any amount of territory and/or power from any dictator without moral repercussions.
2) The aggressor nation in any war is the one that marches over the border, sends bullets over the borders, or otherwise initiates the use of force. Once the gloves are off, the outcome is a matter of ability. This includes any reparations demanded by the victor. One can also presume, reasonably so, that the agressor nation is the one issuing the first declaration of war, unless troops are in conflict first.
Now, on to Mexico.
They willingly let some Americans settle portions of Texas, notably Austin and his group, with the caveat, if I recall, that they become Papists. Since the Texan war of independence is not the Mexican American War, it's only relevance is the inability of the Mexicans to stand up to the agreements they made regarding the borders of the new Republic of Texas. Oh, and they got a little miffed when the independent state of Texas accepted an offer to become one of the sovereign states of the United States. Know what? That wasn't any of their business, no more than the marriage of someone's neighbor to his other neighbor. But is sure pissed off the damn Mexican dictator, didn't it?
Anyhow, the spoiled little children that couldn't accept the loss of Texas a decade before still contested the claim of the Rio Grande as border between Texas and Mexico. Since Mexico was under the control of a strong man who advocated a war with the US, Taylor prudently ordered US troops to the southern extremity of US territory, the Rio Grande, to protect US territory, as it is his duty to do.
Then, of course, it's the curious thing that Mexico declared war on the United States first. Sorta pops the liberal balloon of the US as an agitator and thief, doesn't it? The US was perfectly willing to both continue negotiations and purchase the territory it desired from Mexico, too. Pretty funny to claim that the US stole the land when the Mexicans abandoned talks, overthrew their own leader, and then started the war, wouldn't you say? All you need to do is look at a damn calendar to figure this stuff out.
Should we pool our resources to buy Kandahar a calendar?