If one wishes to analyse the matters of this kind precisely, one needs to separate the meaning of the words involved from the social conventions that govern the settings in which they are typically used.
Typical residents of the UK, US, and culturally similar countries, have some regular interaction with their first cousins, but barely know who their second cousins are, let alone who the still more distant ones are. It is this that makes it reasonable to assume that somebody introduced as 'my cousin' is the speaker's first cousin. It is not that the meaning of the word cousin rules out the possibility of the person's being a second (or third, etc.) cousin; what makes it reasonable to assume that the person is the first cousin is that it would be highly unusual for distant cousins to ever find themselves in a social setting in which one would introduce the other to a third party. If it later turned out that the person is, in fact, the speaker's seventh cousin, one would be surprised, but the surprise would not be of the form 'Then why are you using the word cousin?'; it would rather be expressed as something like 'How come you two are so close if you are only seventh cousins?'.
If one is in a context, such as that of genealogy of the law of inheritance, in which the precise nature of the relationship is important, one would want to specify whether the people one is talking about are first, second, etc. cousins. In casual social interaction, that is, however, usually irrelevant; it is more than enough to know that the two people are somehow related (but not in each other's immediate family). While it is, for the reasons stated above, reasonable to assume that the person introduced as 'my cousin' is the speaker's first cousin, the question whether this is so usually doesn't cross one's mind.
(This answer is an elaboration of the points previously made in Ms Bunting's comments.)