Hey guys, I am sorry for confusion, it was my amazingly stupid error, I forgot to assign replaced value to the original variable vSearchPhrase:=UTF8StringReplace(vSearchPhrase, '"', '\"',[rfReplaceAll,rfIgnoreCase],'');
Bottom line: escaping double quotes with \" works for both Linux and Windows. In both systems, parameter string does not need to be additionally enclosed by quotes!
On Windows, escaping double quotes, by doubling the double quotes "" also works!