I don't think it is about appearance. It is about dialogue and communication. If you want to really talk about things that affect you and your life, these things will quite often be topics you struggle to openly speak about.
I talk about topics that are so hard to talk about that people lie to themselves about it. Things that you don't want to be real are things you try to hide and emotions that don't fit your role in society are part of this.
Becoming vulnerable is the price you have to pay to talk about these kind of topics. And if it happens that you look weak as a side effect you have to pull through it. But this is not about looks, this is about beeing able to interface with the world around you and exposing your true self is the price you have to pay for meaningful conversation.
Vulnerability is being strong and weak at the same time. It's having the strength to know what the end goal is and not get distracted by emotional hurdles while being weak in submitting to authority and peers.
The easiest example of vulnerability I can think of is leaders from other departments asking my opinion on work processes or product ideas. One could argue that's not weakness but I believe it is, in that leaders who believe themselves above everyone else at the company will make policy and standards that ignore important aspects of product development and delivery.
An example of weakness is confessing personal issues to your manager or a peer, so much so they think less of you but generally don't tell you directly. The "vulnerable" version of that is handling your issues as you see them, asking peers and mentors for comments on your work and how you can improve, and simply not talking about them outside that.
I'll admit part of my judgment is shaded by personal bias from my own weaknesses, but that's how I see it.
I prefer to use the term "open" instead of vulnerable, to avoid exactly the situation you're pointing out. And you can be open without being weak or vulnerable if you have self-confidence. Not faked confidence, but the genuine kind that comes from dealing with your shit instead of repressing it and putting on a mask of fake confidence, strength, stoicism, invulnerability, etc.
Yeah you can be open too and this means by definition you are confident about the topic at hand and that you trust the other person enough.
But if all you ever do is only talk about topics that you can confidently speak about with out risking something, then you are missing out on something. And sure you can also go and deal with it on your own till you got the confidence to talk about it without exposing yourself, but there are connections you will be never able to form if all you ever do is deal with yourself.
This is true so long as you appear vulnerable, not weak.