I agree with the prior answer. You need an attorney to assist you. Even if you come to your own conclusion, you will need to be represented in a court proceeding so you might as well get an attorney sooner than later.
Websters is not going to be relevant here. You should look to the will itself to see if it defines "lawful lineal descendants," "descendants" or "children." The langauge of the will controls unless there is ambiguity or the words are susceptible to more than one meaning. You should also look at the Texas Probate Code and case law. Texas Probate Code Section 42 addresses how legitimate children can prove their relationship to the decedent. Texas Probate Code Section 58 addresses will interpretation/construction. Rather than looking at all of this yourself, I recommend getting a consultation with an attorney.
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Yes. There are lots of cases on point but we are not here to do legal research. "Lawful lineal descendants" would mean lineal descendants under Texas law. If paternity can be proven by the child then it is a lineal descendant. This type of heirship litigation requires an experienced probate litigator.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.