COUNTIFS using OR logic on multiple fields

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As one who likes helper columns, I added a column to Table1 that would perform the OR() logic, then used that column in a COUNTIFS() function counting where the OR() logic returns TRUE.

0) I copied the text from the OP and pasted into Excel followed by Text to Columns using space as the delimiter to get the data into Excel. Thus Table1 is in A2:A8, Table2 is in A11:A13, Table3 is in A16:A17.
1) Build the OR() function in column C of Table1.
1a) COUNTIFS($A$11:$A$13,A3) will return 0 when the Field1 text does not exist in Table2 and 1 or more if it does exist. COUNTIFS(...)>0 will therefore return TRUE when Field1 is in Table2, and FALSE when it is not.
1b) A simple boolean comparison will test if the value in Field2 is larger than the value in Table3 =B3>$A$17.
1c) Combine into an OR() function in C3. =OR(COUNTIFS(...)>0,B3>$A$17). Note the mix of relative and absolute references, and copy into C3:C8.
2) The final COUNTIFS() is now simply =COUNTIFS(C3:C8,TRUE)

I know we despise helper columns, but in cases like this, I find that a simple helper column makes the entire task rather simple to program in the spreadsheet.

Edit to add: I expect that the DCOUNT() function could do this without the helper column. It seems to me that, while the Dabs() functions have been around for a long time, no one has ever really come to understand how the criteria range works when you want complex logic. If one spent the time to figure out the right criteria range, I expect DCOUNT() could handle this task.

I took a few minutes to explore DCOUNT() and found that, if I build a criteria range like:

[Field1] -- [Field2]
aaa -- blank
bbb -- blank
blank -- >3

in A11:B14

Then DCOUNT(A2:B8,,A11:B14) would return 5. A quick check using Advanced Filter to see which records this criteria range is identifying, and it appears to be finding the exact same records that my previous formula would find.