Yes; one is an infinitive verb and the other is that same infinitive verb with a reflexive pronoun attached. Levantarte is an example of how you can take the reflexive or object pronoun (in this case, te) and attach it to the end of an infinitive verb rather than put it in front of the conjugated verb.
The two structures that are equivalent here are actually "puedes levantarte" and "te puedes levantar" -- as was pointed out, either one is correct. This works whenever you have a reflexive or direct object pronoun and an infinitive verb.
For example, "Me voy a bañar" (I am going to bathe) can also be written "Voy a bañarme" (moving the me to the infinitive). Or for a non-reflexive one, "Te quiero ayudar" (I want to help you) can be written as "Quiero ayudarte".
That answer is in the database. Either you actually submitted something different, or there was a glitch during submission.
Although it has never happened to me, many other users state that they have entered a matching answer only to be marked as incorrect.
At least you can be reassured you have a valid translation here.
It would be nice if the speaker would pronounce the "s". In a conversation with someone, the personal "you" would be understood. But, in studying and listening, if the "s" is not pronounced, one can write "puede" instead of "puedes", and it would be counted as incorrect. Clear enunciation of words is a systemic problem with Duolingo. It needs to be corrected.
Spanishdict has a good page on how to conjugate reflexive verbs like this: https://www.spanishdict.com/guide/reflexive-verbs-and-reflexive-pronouns
The reflexive conjugations are right there on the page you linked.
Indicative mood, Present tense, Yo conjugation: "me levanto"
"levantarte" is the infinitive with the pronoun added to the end, which can only happen when it is the second in a compound verb phrase. (See the article that Bishop6 posted)