The distributed nature of the Web, as a decentralized system exchanging information between heterogeneous sources, has underlined the need to manage interoperability, i.e., the ability to automatically interpret information in Web documents exchanged between different sources, necessary for efficient information management and search applications. In this context, XML was introduced as a data representation standard that simplifies the tasks of interoperation and integration among heterogeneous data sources, allowing to represent data in (semi-) structured documents consisting of hierarchically nested elements and atomic attributes. However, while XML was shown most effective in exchanging data, i.e., in syntactic interoperability, it has been proven limited when it comes to handling semantics, i.e., semantic interoperability, since it only specifies the syntactic and structural properties of the data without any further semantic meaning. As a result, XML semantic-aware processing has become a motivating challenge in Web data management, requiring dedicated semantic analysis and disambiguation methods to assign well-defined meaning to XML elements and attributes. In this context, most existing approaches: (i) ignore the problem of identifying ambiguous XML elements/nodes, (ii) only partially consider their structural relationships/context, (iii) use syntactic information in processing XML data regardless of the semantics involved, and (iv) are static in adopting fixed disambiguation constraints thus limiting user involvement. In this paper, we provide a new XML Semantic Disambiguation Framework titled XSDF designed to address each of the above limitations, taking as input: an XML document, and then producing as output a semantically augmented XML tree made of unambiguous semantic concepts extracted from a reference machine-readable semantic network. SDF consists of four main modules for: (i) linguistic pre-processing of simple/compound XML node labels and values, (ii) selecting ambiguous XML nodes as targets for disambiguation, (iii) representing target nodes as special sphere neighborhood vectors including all XML structural relationships within a (user-chosen) range, and (iv) running context vectors through a hybrid disambiguation process, combining two approaches: concept-based and context-based disambiguation, allowing the user to tune disambiguation parameters following her needs. Conducted experiments demonstrate the effectiveness and efficiency of our approach in comparison with alternative methods. We also discuss some practical applications of our method, ranging over semantic-aware query rewriting, semantic document clustering and classification, Mobile and Web services search and discovery, as well as blog analysis and event detection in social networks and tweets.