The answer is strongly conditional on discipline and, to a lesser extent, country of origin. Conventions vary widely, as does the degree to which they are institutionalized. For instance, in some fields (e.g., Philosophy), co-authorship is not common and there is no convention about attribution, so absent an explicit note people are are likely to think the more senior author is the primary one. In Sociology, co-authorship is common and the convention is that the first author is the lead author unless there is a note indicating equal authorship. In Economics, co-authorship has become increasingly common over the past few decades but the convention is to list authors alphabetically, regardless of degree of contribution. In some fields the primary author may be determined by looking to see if there's a note specifying to whom correspondence should be directed, regardless of order of authorship on the paper.
Meanwhile in many lab-based science disciplines, where it's sole authorship that's rare, author order is governed by different norms. In some fields, the first author is the one who is primarily responsible for the paper (what that means can vary, too), the last author is the lab head or primary grant-holder, and the order of authors in between is sometimes influenced by other norms. But other conventions exist, too. Knowing what they are and how to interpret them is part of one's socialization into a discipline.
To make things more complicated, some fields—or some journals, or some labs, or some individual authors—may have their own rules or conventions designed to clarify things by listing credit more explicitly. Even worse, there may be a kind of hermeneutics of author-order where people parcel out credit to different contributors regardless of order of authorship, as when someone says "Sure, X is the first author and Y helped him write it up but it's obvious the paper was Z's idea".
In your case, if your lab or unit is using a convention that's not standard in your field the most straightforward solution is to make a note of this in your articles. This isn't an ideal solution because papers will still be cited or referenced without people paying attention to your clarifying note, but there isn't much you can do about that.