This is somewhat of an educated guess, so please point out any inaccuracies.

vivre de la vie [de qqn]: From what I can tell, to live life following a certain model, a model that one cannot necessarily “become.” Hence, from the search results, vivre de la vie de Dieu, vivre de la vie de Jésus, vivre de la vie de Marie, etc. (Consider that when anglophones speak of “living the life of Christ,” they mean living life according to Christian norms and ideals, rather than living as they believed Christ literally lived.)

Note that in your example from Balzac, the full phrase is vivre de la vie des jeunes gens, where la vie des jeunes gens should be parsed as a unit. Hence, your second guess that vivre de is a unit is probably the correct one.

vivre la vie: To live life. Often as vivre la vie de qqc, vivre la vie + adj, etc., meaning to live life in a certain manner. An example from Le Rationalisme de Pascal on Google Books, discussing Pascal’s wager:

… il faut vivre la vie comme si Dieu existait et comme si notre âme était immortelle.

vivre sa vie: To live life as one chooses, roughly. TLFi offers an example from Gide’s Les faux-monnayeurs, where the character has run away from his family home.

Le jeune Bernard a brusquement quitté le foyer familial, où il n’aurait jamais dû entrer. Il est allé « vivre sa vie », comme disait Émile Augier; vivre on ne sait comment, et on ne sait où.

In this form, “de” embodies three relations, not just one, and we can then talk about three locutions.

The idea of a partitive “de” is not to be excluded as a rare possibility, but not in these three instances.
The idea of the prepositional verb does not belong to French grammar, although there is an obvious similarity (change of meaning with change of preposition); the great difference is that in French a complement for the preposition is always needed; if it were otherwise it couldn’t be but exceptional. Anyway, the idea is not relevant in understanding what’s meant: all that matters is the relation of verb to complement that has been embodied in the preposition “de”, that being so on the count of context.

I vivre de la vie des jeunes …

This first case is really an instance of use of the relation found in your archetypal example, “vivre de rapine“. Another such example is “vivre d’amour et d’eau fraiche“; the relation is that of extraction from a given medium of the sustenance for life; in the former that sustenance is purely material, in the latter it is not just a question of a material support for life but also in a figurative way sustenance to the spiritual and intellectual being (here conceived as being love). It might not have been needed to look at this mixed context to show that sustenance can be of a nature other than purely material, but that’s what was intended nevertheless, as well as giving more background. There are no other concepts involved in the case under scrutiny than the second: the sustenance for the spiritual and intellectual life is extracted from the life of the young. However, “life of the young” must be understood only in the context given: there is no material exploitation of this life, no interference with it: the extraction is purely that of concepts that define that life, and that is made through lounging about (flâner); the actual use or “consumption” of these concepts is another matter; in the present development of this idea the author is not particularly versatile since he mentions only a process of emulation (c’est épouser leurs passions).

II vivre de la vie la plus intime

Here the relation is simply “avoir pour”; here are a few instances of use of this relation;

• Ces gens se mentaient du plus pur mensonge. (avaient pour se mentir le plus pur mensonge)
• Elle écrivait de la plus belle écriture qui soit sans indulger dans le trop fait. (elle avait pour écriture l’une des plus belles)

We must understand then “have a life characterised by a close intimacy with others”, that is to say in French, “avoir pour vie une vie des plus intimes”.

III vivre de la vie de Dieu

The third relation, found in a religious context, is used more generally but in a modified form ; apparently, it’s a new usage or a specialised usage proper to the religious establishment; I believe it to be “on the model given by”. It is related to the relation in « II » insofar as in « II » one does not insist upon the model when what’s involved is a mere characteristic.

the passage found here is the key to understanding it ; it is reproduced below;

Se convertir. Nous sentons bien l’exigence profonde de ce que le Christ nous propose. Mais son « non » nous aide. Il dévoile en quoi nous sommes restés « païens », satisfaits par la recherche d’un petit bonheur confortable. L’avertissement de Jésus nous relance vers la vie. Celle qu’il partage. Celle qui vient de Dieu.

“Celle qui vient de Dieu.” means “la vie qui vient de Dieu” ; that is, no doubt, a euphemism for “the model of life according to the christian religion, according to the teachings of God”, in other words “a christian life” or, in French, “une vie de chrétien”.

IV vivre la vie de

This form is used generally when the relation from « III » is used generally (“on the model given by”);

• Il veut vivre la vie de Bohême dans un hotel cinq étoiles.
• Ces gens vivent la vie de cocagne éternellement.
• Ce qu’ils voulaient c’était vivre la vie de famille.

V vivre sa vie

1/ Pour cette locution, the TLFi fournit une définition;

(Syntagme défini) Mener sa vie à sa guise, de manière indépendante, en se libérant des contraintes imposées de l’extérieur.

• Le jeune Bernard a brusquement quitté le foyer familial, où il n’aurait jamais dû entrer. Il est allé « vivre sa vie », comme disait Émile Augier; vivre on ne sait comment, et on ne sait où. (GIDE, Faux-monn., 1925, p. 1118)

2/ A distinction must be made between this locution and those resulting from it by the addition of a complement to “vivre” ; the meaning is then simply that of “vivre”, that of “having a life (of the specified type)”.

• vivre sa vie pleinement — vivre de façon à tirer plein profit de l’existence
  to live your life to the fullest

• vivre sa vie au jour le jour — vivre sans aucun plan, sans préparations
  live from day to day