There are two kinds of electronic signatures: digital signatures and e-signatures. Digital Signatures are based on PKI certificates, e-signatures are supported by some server-based service. Generally, there are several properties that an electronic signature protects: 1. Document integrity: was the document changed since it was signed; 2. Identity: who signed it; 3. Non-repudiation: can the signer later claim that the signature was forged.
The properties that a digital signatures protects depend on the kind of digital certificate you use for signing. If you use a so-called self-signed certificate (Acrobat can create one for you), then a digital signature signed with such certificate protects only the Integrity of the document. If you procure a digital certificate from a reputable Certificate Authority, signature signed with such certificate will protect all three properties: Integrity, Identity and non-Repudiation. The ability to check these properties is contained within the signature itself and the document which it was applied to.
Document Timestamp Signature is a digital signature which is signed with a Timestamp Server. It attests only to the integrity of the document and also to the secure signing time. You need access to a Timestamp server (which usually costs money) in order to use it. Is this what you refer to as "timestamp signature"?
E-signatures' ability to check these properties depends on the abilities of the service that supports them. In order to check these properties for e-signatures you need to go to the Server that supports it and to inquire. At the end of the signing workflow EchoSign can produce a digitally signed (certified) PDF which attests to the validity of e-signatures in this document. Other e-signature services may have similar functionality.
The main difference between the Digital signature and e-Signature is that the verification of a digital signature is self-contained (you only need software that validates it) and for verification of an e-Signature you need to inquire the Service that produced it.