Checking if sequence is almost strictly in ascending order
It does not cover all cases! Consider the sequence [2, 6, 4, 8, 7]. 
Before the return statement, the values of seq1 and seq2 are: $seq_1 = 2$, $seq_2 = 0$. So the function would return $True$. However, the sequence is not almost strictly increasing, because you need to remove at least two elements from it to obtain an increasing sequence. 

So, either the Java implementation is not faithful to the conceptual algorithm, or the conceptual algorithm is wrong. 

The problem can be solved by finding the length of the longest increasing subsequence (LIS) of the given sequence. Call $n$ the length of the original sequence. If the LIS has length $\ge n-1$, the sequence is almost increasing according to your definition. If the LIS has length $\le n-2$, the sequence is not almost increasing.