How Can I Improve Communication with My Boss?

I'm in a bit of a pickle and could really use some advice on talking to my boss at our small M&A firm. Here's the deal:

I've been lucky to have had some good experiences at this place. We're small but mighty, and I work directly with the Director, who used to be at a big bank. But here's the thing: he's not the best at managing me (analyst), and I've never worked so closely with someone at his level before.

The problem is, he's crazy busy. Like meetings all day, every day. And I feel like he sometimes forgets about me or what we talked about, which can get me into trouble.

How can I better keep my Director updated on my progress and tasks? I've tried sending weekly progress reports via email, but they might get buried in his inbox or overlooked amidst his busy schedule.

I've considered scheduling regular check-ins or briefings, but I'm unsure if this is the most effective approach given his packed schedule. 

How can I diplomatically address instances where my Director forgets previous conversations or instructions, leading to misunderstandings or blame? I want to navigate these situations professionally while ensuring clarity and accountability.

I really want to figure out how to communicate better with him. Any tips on keeping him in the loop without bugging him too much? 

Any ideas or advice would be super helpful! Thanks!