During the crafting showcase at E3, when they showed us that we could customize our power armour, the T-45 armour looked exactly the same as the T-60, just with a different colouring. My hypothesis is that the soldiers in the intro are wearing T-45 armour, and T-60 is the Institute's own attempts to improve on the old T-45 design. Perhaps the Institute didn't have any T-51 suits after the war, but had a few examples of T-45. Knowing that they'd need better gear eventually, they took apart the T-45's and began the lengthy process of trying to improve them over the T-51's. Of course, unbeknownst to them, the Enclave would eventually build the X-01 models of Power Armour.
TL;DR? T-60 is the Institute's Post-War advancement on Powered Combat Armour technology.
EDIT: Upon reviewing the footage, it seems that the b designation of T-45 armour looks exactly like the T-60 stuff we're shown. So I think the soldiers are just wearing T-45b suits.