“causing” vs “that ... caused”

Solution 1:

Both are grammatically correct.  What counts as a modifier changes. 

The accident happened on 1st April, causing many casualties. 

Here, the finite verb "happened" forms a clause with "the accident" as its subject.  The participial phrase "causing many accidents" is a supplemental modifier, further describing the content of the entire clause. 

The accident that happened on 1st April caused many casualties. 

Here, the finite verb "happened" forms a clause with "that" as its subject.  The restrictive relative clause "that happened on 1st April" further describes the noun phrase "the accident".  The finite verb "caused" forms a matrix clause, with the entire phrase "the accident that happened on 1st April" as its subject.