Why does "tangent" have multiple meanings that are in conflict with each other?

Perhaps it was just being misused, but the questions crossed my mind before and last night it came up again again.

While talking with someone about job opportunities in NYC, a friend started to talk about cost of living. As we continued to discuss various pros and cons, the friend said to me

Let's get back on the job discussion, cost of living is tangentially related but not important right now.

In this instance, the friend was saying that what we were talking currently (cost of living) has a close relationship with what we started on (working in NYC).

Later in the conversation, our conversation turned towards the amount of time it takes to move from Colorado to NYC. As we talked about the various routes and vehicles for movement, the friend said:

We've gone way off topic — talk about getting lost on a tangent.

In this instance, the tangent is barely related to the original subject.

Is there a misuse of the word tangent here, has tangent just become a word that relates to anything off-topic that has at least the slightest relation to the original topic, or is there something else that I am missing?

Tangentially related question

"Tangent" is a math term that's been picked up by the language at large. It describes a straight line that contacts a circle or curve at exactly one point. It doesn't intersect; it makes contact and then keeps going on the same side of it. On the one hand, two subjects that have one point of contact are "tangentially" related. On the other, once you start going on a tangent, you will keep going and going further away from your original topic and (presumably) never come back to it.

(When the tangent contacts a curve that isn't a circle, it's possible that the tangent may intersect the curve somewhere else, as, for example, in the cited Wikipedia article.)

Tangent does not have conflicting meanings.

In the first example you were talking about jobs in NYC, started talking about something that was tangentially related (cost of living), realised that you were going off on that tangent, and took it back to talking about the job.

In the second example you were talking about jobs in NYC (I think), started talking about something that was tangentially related (moving to NYC), and realised you'd spent a long time on that tangent.

The only difference between the two examples is how far you went down the tangent, not what "tangent" meant.

For a discussion on what "tangent" means, see the answer by Non-Contradiction. But I felt it didn't cover the key point, i.e. that the assertion in the question title was incorrect.

The word tangentially is derived from Latin tangens, which means touching.

In maths, a tangent is a straight line that touches (not intersecting with) a circle or ellipse.

The discussion about the costs of living touches the discussion about job in the sense that they have something in common (in NYC, you earn a lot more money, but life is more expensive). So the Latin word tangens fits here, too, and hence the same English derivative word as in maths is used.