The first two versions don't have a different meaning, the word "own" is just an emphasis.
Now, the sentence
One needs to provide food for himself and his family.
should be avoided. There are people, even linguists who think that's the way to handle a case where you can't or won't tell the gender but "he" does have a gender. Logically speaking a pronoun that serves the purpose to identify a specific gender is not fit to be used as neutral. The formal way is not to replace "one", so only the
One needs to provide food for oneself and one's family.
sentence is correct. Or you can convert the sentence to plural. And speaking of plural, we have the singular they:
One needs to provide food for themself and their family.
A lot of people will jump at this sentence, first because they think "themself" is a non-existent word (my spell checker thinks it too) and second because they think the usage of "they" here is a pronoun-antecedent agreement error. Don't listen to them.
Whenever you need to use a personal pronoun but you can't or won't tell the gender, use "they". It's certainly better than "he", it won't make the sentence unnecessarily complex like "he or she" and of course "he or she" addresses only two genders so you still exclude those who don't identify with either of them—"they" on the other hand now serves as the nonbinary (or gender neutral) pronoun, too.
Of course in formal speech and text this is still frowned upon and teachers who teach English (especially those who teach it as a foreign language) will freak out because of this but still this is the pronoun you're looking for if you want to replace the pronoun "one" or in any other case you don't want to deal with genders. Tom Scott sums it up well in this video and here's some further reading.