It is pretty obvious to me it is speaking about the right of the people of the state of Texas in regards to their state government. Let's go through it.
The faith of the people of Texas stands pledged to the preservation of a republican form of government, and, subject to this limitation only, they have at all times the inalienable right to alter, reform or abolish their government in such manner as they may think expedient.
If it were talking about the Federal government as you read it, it would mean that the people of the State of Texas would have the unilateral right to "alter or reform" the Federal government without any consultation or agreement with the people of other states.
I'm sure you would agree that would be pretty absurd.
Here’s the operative clause of Art 1, Sec 1:
“...the perpetuity of the Union depend upon the preservation of the right of local self-government, unimpaired to all the States.”
I read that as “Screw with local self-government anywhere in the US, with any State, and we have the right to leave.”
The Obama Stimulus Plan abrogates local self-government.