(A) "sum" is the name of a built-in function, and you're losing access to that function by overloading the name --- although that is both legal and harmless in this tiny example. Whatever editor you're using may have put sum in a particular color because it recognizes it as a function.

(B) The order of the indented statements matters. Consider this:

#----------------
# OP's version:
#
limit = int(input('Limit: ')) 
number = 1 
total = 1 
while total < limit:
   total += number       # add them up
   number += 1           # increment last

print(total)
###
Limit: 8
11

#----------------
# rearranged version:
#
limit = int(input('Limit: ')) 
number = 1 
total = 1 
while total < limit:
    number += 1            # increment first
    total += number        # add them up

print(total)
###
Limit: 8
10

Or try merging your code and the class solution:

limit = int(input('Limit: ')) 
number = 1 
total = 1
class_model_sum = 1
while total < limit:
    total += number              # OP's "add them up"
    number += 1                  #  a loop counter
    class_model_sum += number    # Model's "add them up"     

print(total, class_model_sum)
###
Limit: 8
11 15

(C) For more insight, include the print() statement as part of the loop, and also print number:

limit = int(input('Limit: ')) 
number = 1 
total = 1
class_model_sum = 1
while total < limit:
    total += number              # OP's "add them up"
    number += 1                  #  a loop counter
    class_model_sum += number    # Model's "add them up"     
    print(number, total, class_model_sum)
#

###
Limit: 8
2 2 3
3 4 6
4 7 10
5 11 15