In number 1, “being examined” is a gerund phrase (a type of noun phrase), and serves as the subject of the main clause of the sentence. We cannot omit “being”, as it is part of the passive voice. In number 2, “committed (to improving services” is an adjective phrase modifying the subject “we”. We could theoretically add “being” (as a participle, not a gerund), but it’s not required, and English speakers prefer not to use unnecessary words that don’t add value to the communication.