I don’t think they’re inconsistent.
When Malick is explaining HYDRA’s true history to Ward, he says:
HYDRA was founded with the sole purpose of engineering [the Inhuman’s] return. Over generations, HYDRA’s taken different shapes. The entity has been given different names.
HYDRA is not a singular entity; it’s made of many people and has many branches.
The Nazi fork, led by Red Skull, was just one part of a bigger organisation. Malick mentions Red Skull in passing:
Red Skull was following the principle that HYDRA was built upon, that the ultimate power was not of this world. But HYDRA well, we're much, much older than that.
To many people, it would seem like Red Skull was the founder of HYDRA. (And perhaps this was the first time it used that particular name.) This includes Captain America and most of SHIELD – they’ve been keeping their activities with the Monolith very quiet.
It is entirely consistent for this organisation to span multiple centuries, and also for people to erroneously believe it was started by Red Skull.