Looking at the scene in Crisis Core, at the very end when Cloud manages to army crawl up to the dying Zack, and Zack hands over his sword with his brief, heartbreaking "living legacy" speech:
This is not the first time that the Buster has been passed on, especially in this fashion. When Angeal dies - at Zack’s hand, and let’s not get into that mess beyond the painful parallel of how Cloud also feels he’s responsible for Zack’s death even though he had even less choice than Zack did - he passes on the Buster to Zack.
Now, in the English dub (which was the version I played through) Angeal says “Zack, you have my thanks. This is for you. *hands over the sword* Protect your honor, always.” And then Zack has an audio memory of Angeal’s comment at the beginning of the game - Embrace your dreams. If you want to be a hero, you need to have dreams and honor.
Now, let’s compare the English sub (which I’ve watched and found very interesting because it carries a different vibe most of the time):
“You did well Zack. I’m counting on you for the rest. *hands over sword* Never forget your pride.” This is then followed by the same audio memory, but again, with a slightly different vibe - Hold onto your dreams. If you want to be a hero you have to hold onto your dreams and also your pride.
I’m not going to pretend I understand the full significance of the changes between pride/dreams to fit the better cultural perception of the words, but I know enough to say that I do think that moment absolutely influenced how Zack handled that moment with Cloud. Another scene I would say had an effect was the time in Nibelheim (before everything goes wrong) when Zack thanks Cloud for reminding him what the Buster Sword means to him.
I think Zack took the burden of Angeal’s words very seriously (and I say “burden” absolutely referencing when Angeal says that honor can be quite the burden early on plus Gillian’s words on the sword representing the honor of the family.) And I think, as much as he loved Angeal (take that at whatever angle you like, but he did love him in some way,) he probably resented that. I think that was absolutely one of the things he meant when he asked “why is everyone always pushing things on me?” before the last fight with Genesis.
In light of that feeling, I think the last thing Zack intended was to give Cloud the same burden. Looking at Crisis Core only, I would say that I think it was definitely part of what vorpalgirl said: you’re gonna live/you’ll be my living legacy (alternatively, the proof that I existed) is a statement of fact, of the significance of the fact that Cloud is going to live on even though Zack died, that Zack managed to save at least one person he cares about when the others died well before this and even forced him to have a hand in it.
But it’s also motivation for Cloud, because he’s been with Cloud this whole time. He knows that Cloud is struggling (he hasn’t even been conscious until right before Zack left to make his last stand), he knows that Cloud struggled with his self-esteem anyway before all this, and he probably knows that a standard thing that keeps people out of SOLDIER is if they don’t have the willpower/sense of self to make it through the process. That’s a really big thing to ask of a teenager, I remember that much of those years personally. A teenager is still figuring out who the hell they are in that weird not a kid/not an adult half state, this is a big deal and it’s no shock that Cloud “fell short” there - most people probably did.
But Zack is giving Cloud something here, he’s giving him something to focus on because he won’t be able to talk to Cloud all the time or carry him on anymore. All he has left to give are his sword to protect himself, his words to remember and his life so Cloud has a chance at all, and the last one is running out fast.
He never wanted to pass on the burden that Angeal did. He never wanted to push things on Cloud the way they were always pushed on him. But he knew Cloud needed an anchor still, and he couldn’t physically be there to hold his hand anymore, he could barely lift his own blade.
I don’t think he ever imagined it would have ended up that way, that Cloud could even do that, but he did succeed in one thing at least - he gave Cloud a way to hide behind him, until he was ready to stand on his own. And in Advent Children, even though there’s tears in his eyes to have to choose between Sephiroth and Cloud (and he’s lost Sephiroth, don’t get me started on my theories about what Zack was up to in AC) he manages a smile in his voice as he offers Cloud that same bit of strength, that quiet vote of confidence when he’s nearly destroyed again, and helps him back up before he goes.