I feel safe from contradiction if I were to declare that there is no "LGBT manifesto".
The "gender ideology" dialogue tends to refer to the transgender and intersex parts of LGBT+, and not the LGB parts. Nobody reasonably considers intersex people an ideology---these people undeniably exist, although they usually don't advertise being intersex. So I'll focus on the transgender part.
Possibly the most recognized document among transgender people is the WPATH Standards of Care (link), which describes medical best practices in the treatment of transgender people:
The overall goal of the SOC is to provide clinical guidance for health professionals to assist transsexual, transgender, and gender nonconforming people with safe and effective pathways to achieving lasting personal comfort with their gendered selves, in order to maximize their overall health, psychological well-being, and self-fulfillment.
This is written by a large international team of medical professionals who are specialized in the treatment of transgender patients. To my knowledge, most of this team are non-transgender themselves.
This document contains a glossary containing two definitions similar to those listed:
Gender identity: A person’s intrinsic sense of being male (a boy or a man), female (a girl or woman), or an alternative gender (e.g., boygirl, girlboy, transgender, genderqueer, eunuch) (Bockting, 1999; Stoller, 1964).
Gender role or expression: Characteristics in personality, appearance, and behavior that in a given culture and historical period are designated as masculine or feminine (that is, more typical of the male or female social role) (Ruble, Martin, & Berenbaum, 2006). While most individuals present socially in clearly male or female gender roles, some people present in an alternative gender role such as genderqueer or specifically transgender. All people tend to incorporate both masculine and feminine characteristics in their gender expression in varying ways and to varying degrees (Bockting, 2008).
A LGBT+ group GLAAD also has a glossary of terms:
Gender Identity A person's internal, deeply held sense of their gender. For transgender people, their own internal gender identity does not match the sex they were assigned at birth. Most people have a gender identity of man or woman (or boy or girl). For some people, their gender identity does not fit neatly into one of those two choices (see non-binary and/or genderqueer below.) Unlike gender expression (see below) gender identity is not visible to others.
Gender Expression External manifestations of gender, expressed through a person's name, pronouns, clothing, haircut, behavior, voice, and/or body characteristics. Society identifies these cues as masculine and feminine, although what is considered masculine or feminine changes over time and varies by culture. Typically, transgender people seek to align their gender expression with their gender identity, rather than the sex they were assigned at birth.
Sexual Orientation Describes a person's enduring physical, romantic, and/or emotional attraction to another person. Gender identity and sexual orientation are not the same. Transgender people may be straight, lesbian, gay, bisexual, or queer. For example, a person who transitions from male to female and is attracted solely to men would typically identify as a straight woman.
Generally, transgender activists push for political goals that most members of society take for granted: to be treated with basic human dignity in everyday life [school, work, etc.], no sterilization requirements to participate in society, to be able to get married, to use public bathrooms without fear of being attacked, arrested, insulted, etc.
Their opponents tend to regard such political goals as going overboard and potentially dangerous, and thus pejoratively describe these goals as "gender ideology".