Pride flags are kinda loosey goosey across which two pride flags mean which pride-related thing, since there are no consistent rules to it and there are many of them, so you aren't exactly wrong that the first pride flag means that, however, many people just think of it as the flag for gay men.

The new flag (I'm not actually an expert on this history of this so forgive me for being inexact) I believe was created to coincide with a change in strategy, from a focus on just sexual minorities collaberating against oppressors, to a focus on "let's find anyone else the oppressors are targeting and work with them too." Not an effort to get them special treatment, but to signal explicitly that all sexual minorities, gender minorities, racial minorities, feminism, and anyone else can be/should be/are all on the same team and we will have each others back, and stand together against oppressors.

Tld;dr I tend to think of it as "The gangs all here, and you aren't capable of stopping all of us if we work together"