Many have asked questions regarding the different types of signs in use. I have a sheet here from The SEE Center that gives good explanations on the various sign methods commonly in use.
American Sign Language (ASL)
ASL is used by many deaf in the United States, thus its use promotes assimilation into the Deaf Community. ASL is a visual language, and speechreading or listening skills are not needed to learn ASL fluently. Because of its visual nature, ASL is very graphic, and understanding of concepts can be promoted more easily. It has developed over time through usage by deaf individuals and is a free-flowing, natural language. ASL is a language complete in itself. It is not usually written or spoken, but can be translated, just like French or German, to English and vice versa. ASL has it's own syntax and grammar. It does count as a language credit at University level, because it is a separate language.
Pidgin Signed English(PSE) or Signed English
PSE is probably the most widely used communication modality in the United States among deaf and hearing persons who work with them. Many teachers use PSE or Signed English. The vocabulary is drawn from ASL but follows English word order. Words that do not carry information (e.g. to, the, am, etc.) are often dropped, as are the word endings of English (e.g. -ed, -s, -ment, etc.). This means that the signer can easily speak while signing, since it is possible to keep pace with spoken English. It is simpler to learn than ASL or SEE, since one does not need to include all English endings, nor does one to master the structure or idioms of ASL.
Signing Exact English
SEE is based upon signs drawn from ASL and expanded with words, prefixes, tenses, and endings to give a clear and complete visual presentation of English. The ASL sign for the concept of "pretty, lovely, beauty, beautiful" and other such synonyms is retained for beauty, initialized with P for pretty, L for lovely, and the suffix -ful is added for beautiful. The child thus has an opportunity to develop an expanded vocabulary. The learning of this English based sign system may be more comfortable for English-speaking parents. Maximum use of residual hearing and speechreading is encouraged since the signs match the elements of spoken English. SEE encourages the incorporation of ASL features to show intonation visually. SEE does require more signing time that PSE, because of the word endings and prefixes, etc. Overconcentration on signing every word may lead to "colorless" signing.
I think it is fairly easy for a child who knows SEE to code switch to ASL. My friend's little boy, who is 7, is already getting the hang of that. Many school districts use SEE as it does convey the English language well, and facilitates reading, which we know has been a notorious problem with Deaf adults. SEE is universal. PSE can be more heavily influenced by who is doing the signing, ASL can too. Interpreters have more control over what is interpreted in ASL. I know many hearing people become frustrated with that, they think that what they are saying is not being interpreted right, due to lack of knowledge of ASL, i.e., it's the concept of the sentence, not the content.
Deciding which to use or learn, is hard. It is very individual. As we all know which ever path we go down in the communication road with our children it is hard. They all require work from us to learn, and use effectively. I just hope that all of you have gotten the chance to know these options and the others that are available.
NAME WHAT YOU 
No, I am not kidding. My wife who is HH always teaches our kids the proper langauges and good manners.