The two points are not incompatible; the first type of study says that immigrants aren't particularly attracted to countries/states with high welfare benefits, as opposed to (say) high salaries.
The second point says that immigrants tend to use more welfare compared to natives once in a country. Now, I don't know if the second (non-peer-reviewed) [CIS] study is correct on this point. But I'm just saying I don't see any contradiction with the first line of inquiry even if this second finding is true. (You may want to challenge the latter on Skeptics SE.)
Note that a CATO study finds the exact opposite of the CIS study
Overall, immigrants are less likely to consume welfare benefits and, when they do, they generally consume a lower dollar value of benefits than native-born Americans.
Good reasons to be skeptical of such think-tank publications (applies to CIS as well).
I don't have a lot interest in this topic, but on a quick evaluation, CATO uses dollar amounts and CIS the number of people who ever used welfare in some way (it seems). The former measure is probably less misleading.
Actually even using percentage of users (by category of benefits), CATO finds the immigrants use less.

That's a more substantive contradiction with the CIS study if that's you're looking for. With such politically motivated studies (both CIS and CATO's), one needs to read carefully the sampling, inclusion criteria etc., and I don't feel terribly inclined to do that now.
@John has posted a long profile/criticism of CIS; this bit is most relevant:
A September 2015 report by CIS asserted that "immigrant households receive 41 percent more federal welfare than households headed by native-born citizens."[68] The report was criticized on the basis of poor methodology by Alex Nowrasteh of the Cato Institute. Nowrasteh said that the report opted not to examine how much welfare immigrants use, but to examine households led by an immigrant so that the report could count the welfare usage of the immigrant's US-born children, which leads to a misleading estimate of immigrant welfare use.
It turns out that the table snippet you have posted (which is from table 12) is actually titled "Table 12. Use of Means-Tested Programs by Household Head". So Nowrasteh's criticism is directly applicable to this table/snipped, even in the newer CIS report you have linked/quoted (which is dated Oct 2016). Nowrasteh turns out to be a co-author of CATO's own study on the matter, by the way.
Following the breadcrumbs here, Nowraseth has a more detailed explanation why counting by
household gives an inflated use percentage (basically, immigrant families are larger than natives')
Another problem with counting households rather than individuals is that immigrants and natives have different sized households. According to the American Community Survey, immigrant households have on average 3.37 people in them compared to 2.5 people in native-born households. All else remaining equal, we should expect higher welfare use in immigrant households just because they’re larger. CIS should have corrected for household size by focusing on individual welfare use – which is included in the SIPP.
He also gets to an obvious issue I had remarked
The third issue with the CIS report is that they omitted the cash value of welfare benefits consumed by immigrant and native households. CIS only analyzed the use rates for each welfare program but they do not tell you how much welfare was actually consumed. [...] That CIS did not include any information on the monetary value of the benefits received, which is vital to understand the costs and benefits of various welfare programs not to mention fiscal cost estimates, is noteworthy.
@John Doe: yes the CATO study does include illegal immigrants; from its methodology section (page 2), which also helps explain the columns in the tables I quoted above:
We define natives or native-born Americans as those who are
born in the United States, in its territories, or to citizen parents
living abroad. Naturalized Americans are those born abroad
who have since become naturalized U.S. citizens. Noncitizen
immigrants are foreign-born people who are not citizens of the
United States and who include green card holders, refugees,
asylees, temporary migrants, guest workers, and illegal immigrants.
Citizen children of citizen parents includes the children
of both native-born Americans and naturalized immigrants.
Citizen children of noncitizen parents are those who are born in
the United States to foreign-born parents who have not naturalized.
Noncitizen children have not naturalized.
As noted in several places, including the comments below the question and on page 1 of the CATO study [relevant bit quoted below], there are substantial restrictions in benefits that illegal (and legal) immigrants may claim in the US:
Temporary migrants are generally ineligible for welfare benefits.
Lawful permanent residents must wait at least five years
before they are eligible for means-tested welfare benefits, but
states have the option of providing those benefits earlier from
their own tax revenues.
Illegal immigrants are ineligible for
entitlement and means-tested welfare programs apart from
emergency medical care. Naturalized citizens, U.S.-born
children, refugees, and asylees are eligible for all entitlement
and means-tested welfare programs.
These rules have some
exceptions: children of lawful permanent residents are eligible
for SNAP benefits, and states can extend Medicaid benefits
to children and pregnant women regardless of immigration
status. Furthermore, in-kind benefits—such as the National
School Lunch Program; the Special Supplemental Nutrition
Program for Women, Infants, and Children (WIC); and Head
Start—are available regardless of immigration status.