I'm not sure how other contemporary reports are translated, but in those letters and reports from the Indies to the Spanish crown and to others that I've read, the conquistadors who encounter native communities keep claiming that they come in 'friendship' and then proceed to threaten the natives if the latter do not act as 'friends'.
It seems clear that Spanish expectations of this 'friendship' with the natives were, at the very least, that the Spaniards be given tribute and constant logistical support to sustain Spanish communities... which looks a lot like extortion with the threat of violence, i.e. shakedowns for protection money and outright subjugation. It's difficult for me to comprehend how they can mean anything like how I understand the word.
Were the conquistadors being completely and consciously hypocritical in their reports or am I missing something in the context of how they meant the word 'friendship'? How can I understand this? Does this particular usage of 'friendship' have any peculiar baggage from the Spanish experience prior to their overseas conquests?
Ah, allow me to explain. It isn't really a formalized concept in Spanish law as it is a general means of establishing intersocial bonds in many cultures. You can think of the term as more of a temporary to permanent alliance with a few caveats. Whereas the term "alliance" is nebulous, the term "friendship" is easier to grasp cross-culturally. On top of this, the term "friendship" implies a closer familiarity, non-officialness, and overall positive intentions.
Essentially, the Spanish were saying that they arrived among a certain people with no intent to fight or make trouble and are looking for assistance (not as a conquering army) - but they are not opposed to using violence to facilitate their ends. It is not the Spanish necessarily had a positive interest in the communities they encountered but rather that they tried to coerce others into a future relationship that was similar to a friendship between communities.
Your confusion is appropriate and valuable actually - something that the wider public should grasp but doesn't. While our view of the conquest is saturated with stories of cowardly Natives basically surrendering to the Spanish, the sheer ambiguity of terms like "friendship", multiplied over linguistic and cultural barriers, made the intentions of the Spanish unclear and seemingly hypocritical. The Spanish took advantage of this (and by the same token, were taken advantage of in the same way) to the extent that they wouldn't have been able to survive had the communities they encountered understood the totality of what they were saying.
Mind you, different indigenous peoples understood the term in different ways. The subjects of the Aztec and Inca Empires, for example, had dealt with similar types of "extortion" and were under the impression that if they just gave the Spanish what they wanted, they wouldn't really see them again and could more or less continue living as they always had. Others expected something more reciprocal and attacked the Spanish (Coronado's expedition was pretty much one giant example of this). The variety of interpretations and responses one could have to "friendship" with the Spanish does a lot more to explain the Conquest than a whole lot of quasi-racist nonsense about indigenous Americans being inherently meek.
You are correct that the relationship between the Spanish and their friends was lop sided. This doesn't have to do with a weird linguistic meaning attached to friendship. Instead it is more about the language of alliances and of politics. Remember many if the sources you are reading were written by conquistadors to the king. They needed to convince him of their success so that they would be rewarded. By emphasizing friends among the natives they were advancing the success of their efforts. Native allies meant more tribute revenue for the king, more royal subjects, and less fighting. Also, the imbalance between allies is not unique to Spanish colonialism, such relationships were at the heart of Roman expansion too. Especially in conquests outside of the Italian peninsula, Roman legions were heavily supported by allies (alae, where we get our word for ally) recruited among conquered groups or those the willingly submitted to Ronan rule, just like those groups the Spanish called friends.
For the Spanish, check out Matthew Restall's Seven myths the Spanish conquest.