"Can I safely asume that links between subsites on a subdirectories based multisite will be treated as internal links within a single site by Google?"

At best.

Unless they're otherwise straight-up  professional links with zero spam intention, I wouldn't use them.

Don't  worry about all that  header stuff; Google doesn't (very much).

Are your most important words in the topmost locations in your html? That's what you want to care about. Are you  using them in your initial headings? Are you using them in your initial  sentences and paragraphs?

But remember, it's not just about "keywords" it's about supporting/complimentary vocabulary. If you're just throwing your keyword into copy that is otherwise fluff ( contains little topical concepts and terms), you're not going to beat out other sites that do.