If one of my teammate made a mistake in 2018 and it is surfaced now, do I need to prove myself that I did not make that mistake at that time now?
In 2018 I was just a developer but now my client gave me the lead role, so I am responding to all the emails related to that issue.
I am assuming that they may misunderstand that it is my mistake; do I need to explicitly mention that I did not do the mistake to do not lose confidence on me? The person who made the mistake already left.
If you are now the lead, then you own the responsibility for everything that happened, happens, and will happen. Blaming someone who has left will have unfavorable optics about your character to both your customer and your team...even if it is valid. Moreover, that person was a member of a team and so, if he made a mistake, the team made a mistake. Everyone succeeds or everyone fails.
You do need to identify the root cause and have a plan to fix it so it does not happen in the future but you need to report the root cause in a way that does not blame one individual but rather blames the team's processes at that time.
In all cases, own it and now fix it. You will be far more respected in the end.
I am assuming that they may misunderstand that it is my mistake
If you are clearly speaking for your team as a leader and not just yourself, it would be very hard to misunderstand. You should be talking about "we", "our mistake" and "we will fix it". It doesn't matter and quite frankly is none of the clients business, who exactly on the team made the mistake.
In my opinion, you should first start by acknowledging the issue and clearly communicating the plan of action to resolve this issue. You should accept that the mistake was done by the team and do not name the person who made the mistake. Own it as a team mistake, move forward to resolve the issue, and gain your client's trust.
