Phrased a little differently "Does a seller really care if it is cash offer or an offer with a reasonable down payment with financing?"
Cash is cash - period. While the amount offered is a significant issue its specific source is not usually a significant point.
However, a seller may look at a cash offer as one less hurdle for the buyer to jump over - no financing contingency. [Most real estate offers come with a couple of contingencies - an inspection and financing.]
The answer to the advantage gained by a cash offer, in my opinion, depends on the individual property, type of sale (short, estate) and market price.
If the property is a rehab project or has significant issues, financing may be a non-issue. Lenders have requirements concerning habitability, etc. that must be met before a loan will be approved. [There are some loans that allow the inclusion of repair costs in the loan. Speak to a lender about these FHA loans.].
If the subject property is located in a condo community that does not qualify for financing then cash is likely an only option. [Or perhaps owner financing]
If the sale is an estate sale you may find sellers trying to liquidate and a cash offer may be viewed more favorably.
Finally in a situation where the property is very desirable (priced right, good location) cash may gain the buyer leverage. These are those situations I refer to as a multiple offer situations - a good place for a seller to find themselves in!
If you are considering buying a home in Brevard County, Florida, give me a call if I can assist in any way.