The two senses you mention are adverbial. In sense 1,
as long as you live,
as long as expresses an 'amount of relative duration'. In sense 2,
as long as I can arrive on time,
as long as expresses 'on the condition that', 'provided that', or 'if only'.
These senses are summed up by definition 1b of long, adv.1 in OED Online:
In the comparative and superlative, or preceded by advs. of comparison (as, how, so, thus, too, etc.), the adv. indicates amount of relative duration. (Cf. long adj.1 8a.) so (or as) long as: often nearly equivalent to ‘provided that’, ‘if only’. Also, long as, ellipt. for so (or as) long as.
c900 tr. Bede Eccl. Hist. iv.xxv. (Schipper) 496 Ic..þe..ætywde..hu lange þu on hreowe awunian sceole.
971 Blickl. Hom. 169 Swa lange swa ge ðisdydon ðara anum ðe on me gelyfdon.
["long, adv.1". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/109979?redirectedFrom=as+long+as (accessed February 18, 2016). Bold emphasis mine.]
In the first quote shown, from Bede's Ecclesiastical History, which dates to sometime before 950, "hu lange" translates as 'how long' (Latin quamdiu, see the fourth bullet at the Bosworth-Toller Anglo-Saxon Dictionary entry for 'hreów': quamdiu pœnitentiæ insistere, which may be translated as "how long [a time] to continue your penance"--translation from Saint Bede, The Complete Works of Venerable Bede, in the original Latin, collated with the Manuscripts, and various printed editions, and accompanied by a new English translation of the Historical Works, and a Life of the Author).
In the second quote shown, from Blickling Homilies, which dates to 971, "Swa lange swa" translates literally to 'as long as', used in your second sense, that is, it means 'provided that' or, more precisely in this case, 'on account of the fact that':
...as long as you did this to only one who believed in me...
(From Blickling Homilies: Edition and Translation, Richard J. Kelly,
Bloomsbury Publishing, Jul 15, 2010, p. 119.)
Image from the untranslated page:
(From The Blickling Homilies of the Tenth Century from the Marquis of Lothian's Unique Ms. A.D. 971, Volume 1, Richard Morris,
Early English text society, 1880, p. 169. See p. 168 for the translation.)
- The OED lexicographers did not consider the adverbial senses of 'as long as' indicating "amount of relative duration" and "nearly equivalent to 'provided that', 'if only'" sufficiently distinct to warrant separate definitions or separate lists of supporting attestations.
- Attestation from approximately the same period illustrates both adverbial senses.
Additionally, the etymology given for these adverbial senses of 'long' (adv.1, definition 1b) suggests a derivation from adjectival uses of 'long' (adj.1). The earliest attestations provided for the adverbial senses and the adjectival senses are, however, from the same work, dated to around 888 (Ælfred tr. Boethius, De Consol. Philosophy).
The cross-reference in the OED adverb definition 1b to definition 8a of the adjective for "amount of relative duration", but not for 'provided that', does suggest the lexicographers see a closer relationship between that sense of 'amount of relative duration' and the sense of the etymological source adjective.
The cross-referenced adjectival sense of 'long' is this:
Having (more or less, or a specified) extension serially or temporally.
["long, adj.1 and n.". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/109975 (accessed February 18, 2016).]
The evidence presented, however, does not support a conclusion that one of the adverbial senses was the source or "original" form of the other adverbial sense.
It is clear that the adverbial sense of 'long' in the phrase 'as long as', meaning 'amount of relative duration' refers to temporal extension. That is not clear in the case of the adverbial sense "nearly equivalent to 'provided that'". For 'provided that' and similar meanings, I prefer logical and thus serial extension to temporal extension. In this context, serial refers to "...forming part of...a series, in respect...of conceptual order" (OED Online, serial, adj. and n., emphasis mine).
Thus, while the base meaning of 'long' may be the same in both cases, that is, it conveys the idea of extension, in the first case, (your sense 1) 'long' refers to a temporal extension, and in the second case (your sense 2) 'long' refers to a serial extension.