I wrote an answer regarding the background of the Texas Revolution here.

Regarding the nature of Texas, there's actually a pretty simple answer. Even at the time, people were not exactly sure what to do with it. Two parties ended up forming, which I will, for convenience, call the Democrats* and the Nationalists.

The party headed by Sam Houston was broadly in favor of annexation into the United States. There was an early attempt at this post-independence as Houston became like the George Washington of Texas history, but the US did not allow it to carry through. Particularly, it was not allowed because of the ongoing Slave Debate rocking the nation, and Texas would throw significant power to one side that neither side in the debate was willing to risk at the time. The formal application for annexation drafted in 1837 was denied on these grounds.

So, in the following administration of Mirabeau B. Lamar, annexation attempts ceased. Lamar's nationalist stance sought the recognition of Texas as an independent republic by leading international powers, and the still-present Texan embassies in London and Paris are reflective of this desire.

Lamar disagreed with Houston on many other fronts, such as while Houston was an adopted Cherokee, Lamar advocated the removal of Cherokee and Comanche peoples outside Texan borders. Lamar was also interested in government-funded public education, and the creation of a National Bank. Houston sought to normalize relations with Mexico and the United States, though he also curried favor with European powers at the same time.

Lamar's government was a failure in many ways. He started costly wars against indigenous peoples, nearly doubled spending, and caused mass inflation by printing worthless paper money to try to fill his dream of a treasury. In this instance, it basically bankrupted the country, unlike Hamilton's work over in America. On the subject of slavery, Houston was something of a moderate - he opposed extending slavery further while not necessarily calling for abolition, but rather he, like Lincoln, favored Unionism over either stance particularly. Houston would eventually break from the Democrats over the Kansas-Nebraska act and his fear that more indigenous peoples would be disenfranchised and robbed of land, he would oppose the secession of Texas from the US and, when it did happen, he opposed the reannexation of Texas into the Confederate States - which he viewed as something of an abomination. He died in 1863, after having officially freed his slaves, supposedly after having read them the Emancipation Proclamation. It is said that they largely chose to stay with Houston. One of these former slaves, Joshua, took Houston's surname, and became a successful businessman and local politician. Sam had taught him how to read and write, and gave him a true education. It is said that after Sam's death, his widow moved away and fell into financial trouble - where Joshua offered her $2000 (~$40,000 today) due to how close he felt to the family. I emphasize all these points to demonstrate the complexity of Sam Houston on this particular issue.

So, with this context, I suppose you can see how lines were drawn. Houston and his friends hoped for a quick annexation into the US that just didn't come, and Lamar and his clique sought raw independence. Although annexation was widely popular at first, even Houston's group started preparing for the eventuality of a longer-lasting country than they had hoped. While at first everyone hoped to be annexed, they began to acknowledge the potential of that annexation simply not coming, and prepared for it - until, about a decade after the Revolution, it did happen as one American president's passion project.

The circumstances leading into the preference for annexation versus independence was highly politicized and the characters of the men involved should tell you what you need to know about who sat where, particularly. It's not quite so easy to separate into "Conservative" and "Liberal" groups per se, but the right idea is there in that dichotomy. The "Conservatives" sought true and lasting independence, while the "Liberals" prepared for lasting independence while hoping for annexation.

In fact, one of the reasons the United States eventually did annex Texas was due to the fear that it would abandon slavery if left to its own devices, which many in the South feared would be a domino to topple their own enterprises. A bit ironic how the people Lamar liked best came around against him like that.

*Although leaders of both parties were technically aligned with the Democrats traditionally, I will call Houston's party the "Democrats" just out of convention, and perhaps to emphasize his American connections