Yes, they are both grammatical, and have very similar meanings, even though their structures are radically different. (Two minor corrections, which don't affect what you are asking about: "walked in his direction", and the idiom is for dear life. Alternatively, you could say "for his life")
In "On seeing that ... , he turned", the initial clause is a prepositional phrase locating the action in time. It is functionally equivalent to "After lunch" or "at three o'clock". It modifies the whole predicate.
In the second sentence, "Seeing that ..., he turned", the initial clause is either an adjectival clause modifying "he", or possibly an absolute clause modifying the whole sentence. But it may be doing more than specifying the time - it might, for example, imply causation or motivation.
The other three sentences are a bit more complicated. I find them all a bit unnatural. In the first one, "He got up from his seat, and walked towards her, seeing that she had arrived", I find it so unnatural in this sense that for me a different, idiomatic, use of 'seeing that' appears: this means 'even though', or 'because'. So the primary reading of that to me is something like "Because she had arrived, he got up ...", with an implication that he didn't want to, or wasn't ready to, but felt obliged to because she had made the effort, or something like that.
The last two are fine, but as I say they feel a bit unnatural to me. I find the last one, with the comma, slightly better.