If I attend Mass on 9 consecutive first Fridays, and receive Holy Communion (hopefully in the state of grace), and/or:

1) Attended said Masses specifically to avail myself of the attached promises, but forgot to form that specific/explicit/conscious intention at one or more of the Masses;

Or

2) Not being sure of what, precisely, the formed intention is supposed to be (eg., sometimes I only had the intention to “attend and receive Communion for the nine first Friday’s and the promises attached,” or some such conscious intention, but at other first Friday Masses, I wondered if the conscious intention required a more specific formula, making reference to “atoning for the outrages against the sacred heart (meaning my intentions, implicit or explicit in #1 above did not suffice).”

Today would be my 9th first Friday Mass, and I’m not sure if I qualified for the promises, because of this issue about the requisite intention.

What do you think?

Clearly you've gone with the intent to do the First Fridays, so it seems to me that your intention "qualifies"

I asked the same questions to an SSPX priest, and this is his response (shared in case any are scripting on this point):

“Remember we should do the first Fridays as an ongoing devotion not just limited to 9.
If we've made the intention of doing the devotion at the beginning that intention carries through the whole exercise/devotion unless we renounce that intention or announce something contrary to it.
I think we should receive the benefit from the devotion.”