The present participle is what we call it in England. You may, if you wish, try to divide its usages into gerund, gerundive, verbal participle or whatever. I mostly find that exercise pointless and prefer to distinguish between nouns and verbs.
So the question here becomes, which is it better to use:
noun + "of" + noun (e.g. the sponsorship of our mission... the sponsoring of our mission...) ?
verb + noun (e.g. sponsoring our mission)?
My answer is that, for reasons of economy, style and simplicity, number 2 is usually preferable to number 1.
There are some grammar terms that English people veryveryrarely use, like gerund, aorist and preterite. These occur once in every three million printed words. The vast majority of native English speakers have no idea what they mean.
And yet we manage to use words that end in -ing in all the ways your book describes; we just do it, without having a name for it. We can even use verbs, nouns and adjectives without necessarily knowing those technical terms!
The way I understand it is that the present-participle form (words in -ing) can be nouns, adjectives or verbs. That's quite enough, thank you.