There is not as I am aware a precise or universal definition of 'anchor.'
But, in general, a district's anchor has the ability to control or substantially influence the outcome of elections because it contains the majority or a substantial plurality of the district's population.
There are occasionally districts that have two large population centers (TX-28) is an example, but that is relatively rare.
I will say people usually talk about a county serving as anchor, so the Austin claim is a little differently worded. It works for Travis County because Austin is so dominant, but just know that the data breakouts are usually reported by county.
As for other cities, Texas, of course, has three of the cities larger than Austin.
Dallas has CD-30 (Eddie Bernice Johnson) making up a good part of the city of Dallas (it is wholly based in the county).
Nearly all the people in CD-32 (Sessions) and half the population of CD-24 (Marchant) also live in Dallas County - though not necessarily all in the city of Dallas.
In San Antonio, CD-20 (Castro) is nearly all contained in the city of San Antonio.
In Houston, you have CD-9 (Al Green) almost all in Harris County and substantially in Houston. In addition, CD-18 (Jackson Lee) and CD-29 (Gene Green) are wholly in Harris and largely in Houston.