Some trans people (so-called "binary" trans people) do indeed transition to being male or being female. Other trans people present their gender as neither female nor male but somewhere in the middle, and they are called non-binary. So both types of trans people do exist.
The "there are only two genders" people, in my view, are trying to invalidate all trans people because the statement is usually accompanied by, "and they're determined by your chromosomes." Even without that second part (which is a rare case in my estimation), the statement attempts to erase non-binary trans people, which other (binary) trans people do not agree with.
For someone who doesn't know much about gender identity, you are hitting on a relevant and interesting issue for trans people: the slightly conflicting desires of binary and non-binary trans people. A binary trans person will often want to embrace the stereotypes and norms of their gender as a tool to help them be seen as the gender that they identify with. This means, for women, wearing things like makeup and dresses, and for men, growing a beard and dressing in more masculine ways. However, non-binary trans people are in a sense defying and blending gender norms to reinforce that gender is a spectrum, not a binary. While both groups of trans people generally have similar views on gender (I think nearly all would say that there are more than two genders), in their personal lives they tend to skew in two different directions.
This is also relevant when it comes to gender pronouns. Non-binary trans people often advocate for everyone to share preferred gender pronouns rather than assuming, while some binary trans people would prefer to have their pronouns assumed, so long as the assumption is correct, because such an assumption validates their gender identity. There was a recent controversy where Natalie Wynn (aka ContraPoints), a binary trans woman, expressed her preference for gender pronouns to be assumed while acknowledging and supporting non-binary trans people's desire for no assumptions, and she was widely criticized (wrongly, in my view) for this.