Do what you're more comfortable doing!
Offer what you think the house is worth!
The only way to tell what a "better offer" is.........is to see if the seller accepts it!
Really - who cares what we think - we're not selling you the house! The seller's situation and motivation may dictate what is more appealing to him or her.
Now, if you're talking about all cash compared to, for example, an FHA (3.5% down) loan, then it's a different story, and may play out differently, but with 40-60 % down, that's a strong offer in its own right, and most sellers would be happy with those terms (assuming there is a pre approval to go along with it).
By the way - you didn't mention whether you would be putting an "appraisal" contingency in your all cash offer!