Familiaris does indeed indicate a deeper, more intimate level of friendship. Pliny the Younger often uses it thus. In fact, the letters of Pliny offer a wealth of evidence relevant to this topic (therefore, he'll figure quite prominently in this answer).
In the following sentence from letter 4.17, he distinguishes regular friendship (amicitia) from intimate friendship (amicitia familiaris = familiaritas):
est quidem mihi cum isto, contra quem me advocas, non plane familiaris sed tamen amicitia.
I have with that man (C. Caecilius) against whom you ask me to serve as opposing counsel, obviously not intimite friendship, true, but friendship nonetheless.
Contubernalis can also denote an intimate friend – a bosom buddy, boon companion, mate, or the like.
This example (letter 4.14) contrasts contubernalis and amicus:
idem C. Caluisium, contubernalem meum amicum tuum, arta propinquitate complectitur; est enim filius sororis.
This same man (Varisidius Nepos) also has a very close relationship with C. Calvisius, my mate and your friend; for he's his nephew.
This example is from letter 2.13. I include it mainly because it also includes the adverbial form of familiaris, familiariter, which may indicate that familiaris and contubernalis involve similar levels of intimacy:
hunc ego, cum simul studeremus, arte familiariterque dilexi; ille meus in urbe ille in secessu contubernalis, cum hoc seria cum hoc iocos miscui.
When we were students together, I had a close and intimate affection for him (Voconius Romanus); he has been my mate both in Rome and in retirement, and with him I have shared seriousness and levity.
There's also sodalis. This example is also from letter 2.13:
quid enim illo aut fidelius amico aut sodale iucundius?
For what could be more trusty/reliable/loyal than he as a friend or more pleasant than he as a mate?
This passage may contain a hint that one expects different things from an amicus and a sodalis: an amicus is someone you can rely on, whereas a sodalis is someone you want to hang out with. And that makes sense, given that amicus and amicitia were used also for things such as client/patron relationships that might not have involved any particularly warm feeling on one or both sides, but did involve obligations. (As an extension, it was also used of the relationship between Rome and foreign kings, such as the Numidian king Micipsa in Sallust, Bellum Jugurthinum.) Still, I wouldn't want to press that point too far based on one passage.
For a shallower level of friendship, such as a casual acquaintance, one could refer to the person as notus mihi, 'known to me,' or as Horace says in Satire 1.9, notus mihi nomine tantum, 'known to me by name only':
ibam forte via sacra, sicut meus est mos,
nescio quid meditans nugarum, totus in illis:
accurrit quidam notus mihi nomine tantum
arreptaque manu 'quid agis, dulcissime rerum?'
By chance I was going along the Via Sacra, as is my wont, mulling over some sort of trifles, totally absorbed in them: some guy known to me by name only ran up, grasped my hand, and said, 'How are you, sweetest of things?'
Pliny also uses (to refer to himself) homo minime familiaris, 'a man not at all on intimate terms' in letter 2.6. Given the subject of the letter (a man who serves different food and drink to his guests at dinner parties, to rank them), Pliny is using this to put himself at a level that's not just less than familiaris but significantly below amicus:
longum est altius repetere nec refert, quemadmodum acciderit, ut homo minime familiaris cenarem apud quendam, ut sibi uidebatur, lautum et diligentem, ut mihi, sordidum simul et sumptuosum.
It would take too long to go into the back story, and it doesn't much matter any way, how it came about that I, a man not at all on intimate terms with him, was dining at the house of someone who was, at it seemed to him, elegant and yet economical, but as it seemed to me, vulgar and extravagant at the same time.
