What kind of sign language is it when you do not sign pure ASL with its own grammar, or use signed exact english? It is called pidgin sign(ed) english, or PSE. A more recent term is "contact signing." This is the form of sign language I have seen most commonly used, especially at Gallaudet and NTID.
What is Pidgin Sign(ed) English?
PSE is not a true language and lacks rules. It is viewed by sign linguistics experts as a way to "bridge" the gap between native ASL speakers and native English speakers. Native speakers can be either deaf or hearing. It contains a mix of ASL rules and English grammar. The signs used in PSE come from ASL, but they are not used in an ASL-ish way, but rather in a more normal English pattern.
PSE speakers also may not utilize certain elements of the English language such as the words "the," etc., to speed up communication. Some PSE users like myself, do sometimes use elements like "the" or "am." One thing I don't use, is the endings of words - I don't sign "ing," or don't always sign or fingerspell the past tense. For example, I might say "I finish clean" instead of "I cleaned." PSE is quite individualistic and users communicate in whatever way they feel comfortable with.
Research into Pidgin Signed English
Ceil Lucas of Gallaudet University's Department of ASL, Linguistics, and Interpretation has done a fair amount of research into PSE together with Clayton Valli. Volume 1, number 1 (2001), pages 145-152 of the Spanish journal Estudios de Sociolinguistica published his article, "Language contact phenomena in deaf communities." Pages 149 to 151 describe a project on contact signing that examined the nature of PSE. At the end of the article, there is a short bibliography that includes resources such as an article by J. Woodward, "Some characteristics of Pidgin Sign English" in Sign Language Studies (vol. 3, pp. 39-46, 1973).Lucas' article can be downloaded in PDF form on the website of Estudios de Sociolinguistica. Lucas and Valli's work is in more detail in the 1992 book Language Contact in the American Deaf Community (ISBN 0-12-458040-8) (compare prices).
Additional Bibliographic Resources
"Interpreter's effectiveness in reverse interpreting: Pidgin Signed English and American Sign Language."This University of Rochester dissertation by Tracy Hurwitz is available from ProQuest Information and Learning.
Barbara Luetke-Stahlmanwrote the article "Three PSE studies: Implications for educators," In M.P. Moeller (Ed.), Proceedings: Issues in Language and Deafness (1993). Omaha, NE: Boys Town National Research Hospital.
There is also some discussion of PSE in the article "Does ASL Teach English."