All of these work for what you are looking for. A strap is the longer piece of the purse that goes over your shoulder or across your body. Where a handle is the small part found on some purses that are on top of the bag.
Native speakers will say like this: “My bag handle tore!”.
There are lots of other ways to say it, but this one is the most accurate and most commonly used.
Here is an example on how to use this in a conversation:
A: My bag handle tore! I can't use it in this state.
B: Let me see if I can fix it.
ネイティブスピーカーなら、このように言います：
“My bag handle tore!”（バッグの持ち手がとれた）
他にもいろいろな言い方ができますが、これが最も正確で一般的です。
会話で使うと次のようになります：
A: My bag handle tore! I can't use it in this state.
［訳］バッグの持ち手がとれた！これじゃ使えない
B: Let me see if I can fix it.
［訳］直せるかちょっと見せて
With handbags there are usually handles so that you may carry the bag in your hands, or sometimes there is a shoulder strap so that the bag may be slung over your shoulders. Sometimes bags have both straps and handles.
"My handbag is being repaired as one of the handles broke."
ハンドバッグには普通手で持つための「handle (持ち手)」が付いています。肩から下げるための「shoulder strap (ショルダーストラップ)」が付いていることもあります。「handle」と「strap」両方あるかばんもあります。
"My handbag is being repaired as one of the handles broke."
(ハンドバッグの handle が壊れたので、修理に出しています)
You could call the part of a bag which you hold a bag handle, or simply a handle. This comes from the word 'hand' because you hold the handle in your hand.
The part that people use to handle a bag is called a "bag handle" or a "handle of a bag". For example:
-My bag handles broke and I want to go get them fixed.
-The handles of these bags are hurting my hands because the bag is very heavy
カバンの「持ち手（the part that people use to handle a bag）」は "bag handle" や "handle of a bag" と言います。
例：
-My bag handles broke and I want to go get them fixed.
カバンの持ち手が壊れたから、直しに行きたい。
-The handles of these bags are hurting my hands because the bag is very heavy.
カバンがすごく重いから、（持ち手で）手が痛い。
We call the part ”handle”. It is a small piece attached on top of a bag.
e.g. The handle of my bag is broken. I will take it to the shop to see if it can get repaired. カバンの持ち手が壊れた。修理できるかどうかお店に持って行ってみる。
Bytheway, the word in Japanese English ”ハンドル” does not refer to the part with which you drive a vehicle. In English, they call it ”steerling wheel”. This will be good to remember, since they will not get what you mean if you said ”the handle is heavy” as in Japanese to a Native speaker. Correctly, you would say ”This steering wheel feels heavy”. They may understand the Japanese English word “ハンドル” if they lived in Japan and have some experience learning the meaning of the word though!
Hope this is helpful information.
カナダ 
日本 
イギリス 
アメリカ合衆国 
南アフリカ 
