So, It seems that there are different types of enlightenment. At least two: enlightenment and right enlightenment.
There are several different types and styles of enlightenment: Arhat, Pratyekabuddha, Bodhisattva, and Buddha. There are several "levels" to the bodhsiattva (10) and buddha (3) attainments. I'm not sure that Arhat and Pratyekabuddha have levels of attainment.
I think that Dalai Lama confronted with this would correct himself right away.
No, I think he would stand by the statement, which if quoted correctly in the start of this thread was,
"Enlightenment just for oneself isn't right enlightenment according to mahayana view."
The "according to the Mahayana view" is the context that makes the statement correct.
From Gompopa's "Jewel Ornament of Liberation":
"The Hearer (Arhat) and Solitary Realizer (Pratyekabuddha) families are inferior by virtue of the fact that they fully purify their families by dispelling only the obscuration of afflicting emotions. The Mahayana is superior because it fully purifies its family by dispelling two obscurations--afflicting emotions and the subtle obscurations to enlightenment. Therefore, the Mahayana family is superior and unsurpassed.
Gompopa goes on to say that the Buddha "encourages" Arhats and Pratyekabuddhas to attain Buddhahood.
"...light radiates through the Buddha's wisdom and touches the mental bodies of the Hearers and Solitary Realizers. As soon as the light reaches them, they arise from their unafflicted meditations.
" He then goes on to tell them that their experience of nirvana is not the final nirvana, and that they should work towards the realization of the Buddha.
Of course that is terribly politically incorrect to say on a forum that caters to Theravadans.
Such sentences shouldn't be taken outside the whole teaching.