The second cited source (Wade et al) looks a good read but I would not trust it implicitly. Lumping all South Asian influences on Southeast Asia in to a single text (leaving out China, Tibet, Iran, the Middle East and Africa) seems an exercise in artificial scope.
Remember that 'Southeast Asia' itself is a vague distinction, particularly in the north. Half of Vietnam was Hindu until the 15th century, Burma never really existed (and to some extent still doesn't), similarly direct contact from the Pala Kingdom with Nanzhao is well evidenced by surviving images, documents and unmistakably Hindu carvings well within the borders of what is now modern China. Then there are each of the attested or probable seafaring dynasties (eg. Chera, Chola, Gupta, Nanda, Pallava, Pandya) and their various activities and records, in some cases many of which still exist, the establishment of kingdoms or fiefdoms of varying longevity (influentially Funan, but also in Burma, Thailand, Malaysia and Indonesia), comments on those from visitors (many in classical Chinese), and the sum of archaeological work.
Remember also, quite apart from the textual evidence, that mtDNA evidence now points to the peoples of significant parts of coastal China (eg. Shanghai region) having migrated north from Southeast Asia and not really being 'Chinese' at all, and irrefutable linguistic evidence shows Madagascar was settled by Austronesians who presumably sailed all the way from Southeast Asia. What all this means is that the period of recorded history does not intersect very much with much of the early period of cultural and linguistic exchange, therefore take everything you read with a grain of salt.
Remember that 'Southeast Asia' itself is a vague distinction, particularly in the north. Half of Vietnam was Hindu until the 15th century, Burma never really existed (and to some extent still doesn't), similarly direct contact from the Pala Kingdom with Nanzhao is well evidenced by surviving images, documents and unmistakably Hindu carvings well within the borders of what is now modern China. Then there are each of the attested or probable seafaring dynasties (eg. Chera, Chola, Gupta, Nanda, Pallava, Pandya) and their various activities and records, in some cases many of which still exist, the establishment of kingdoms or fiefdoms of varying longevity (influentially Funan, but also in Burma, Thailand, Malaysia and Indonesia), comments on those from visitors (many in classical Chinese), and the sum of archaeological work.
Remember also, quite apart from the textual evidence, that mtDNA evidence now points to the peoples of significant parts of coastal China (eg. Shanghai region) having migrated north from Southeast Asia and not really being 'Chinese' at all, and irrefutable linguistic evidence shows Madagascar was settled by Austronesians who presumably sailed all the way from Southeast Asia. What all this means is that the period of recorded history does not intersect very much with much of the early period of cultural and linguistic exchange, therefore take everything you read with a grain of salt.