This was my first time seeing a teacher cry.
This was the first time I'd seen a teacher cry.
Question 1: Is there any difference between the two?
Question 2: Which is more common/sounds more natural?
Both are Correct , both are Common , both are Natural !
Here is a slight subtle Difference , which may or may not be intentional :
My team of Academic Evaluators had seen a lot & most of them had seen teachers cry, but ....
This was my first time seeing a teacher cry.
I had heard of teachers crying when Academic Evaluators ask tough questions, but ....
This was the first time I'd seen a teacher cry.
Point is not to worry too much about the "better" or "best" way , in this case.
In terms of syntax, I think it is more common to use relative clauses (as in "first time that I'd seen a teacher cry") in spoken English than written. In written English, however, non-finite verb phrases (a participle phrase in this case, as in "first time seeing a teacher cry") may be preferred to a relative clause for its conciseness(reduced word count and fewer finite verbs).