We use "si" when we are referring to the name of a person as the person himself/herself.
"Si Luke at si lukey123 ay iisang tao" = Luke and lukey123 are just one (one and the same) person. - In this sentence, Luke is a person and so is lukey123. Hence, we use "si".
( Also, in this case, it is be better to use "si" twice than a single "sina" to give emphasis on each name. )
"Luke ang tunay na pangalan ng miyembro na si lukey123" = Luke is the real name of the member that is lukey123 - In this sentence, only lukey123 is the person, while Luke is just a name. Hence, "si" is used only with "lukey123", but not with "Luke".
Had your sentence been, "I am Luke or lukey123", the Tagalog of that would be, "Ako ay si Luke o si lukey123". You would need the "si" for the names because you're saying that "Luke" or "lukey123" is you, the person. However, if your sentence is, "My name is Luke or lukey123," the sentence itself says that "Luke" and "lukey123" are just names. Therefore, you don't need the "si".
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