No - the sim card only provides an ICCID number to the phone. That number is read once per boot as the phone just uses the sim to identify what account to bill for service when registering with the nearby cell towers (IMEI and ICCID together let the system know what number to assign your phone and who to bill).
Something else must be causing your observed behavior.
Keep in mind, there is no guarantee two phones in the same location are not on distinct cell towers. Especially if one tower is overloaded or doesn't have high speed data so the phone /carrier might pick a faster connection that happens to have less bars of signal.
What does your carrier say if you call them for a site survey / diagnostic?
I think I found the Answer I was looking for in How U.S. Carriers Fool You Into Thinking Your iPhone 4S Is Unlocked
Short answer "You can't do it!" If your phone is CDMA, its permanently locked to its CDMA network of origin.
Unlocking your Iphone 4S
Among the three major carriers operating in the U.S., only one, AT&T, will not unlock any iPhone. Because I purchased a Verizon-designated handset, the following information relates to those customers that purchase Verizon iPhones, but the technology and result is the same for any carrier.
When I called Verizon tech support to unlock my phone prior to leaving the U.S., I was told that I had to have a foreign SIM card installed in order to complete the process and that the phone would have to log in to the Verizon network on WiFi. The reason for this procedure is that the phone must validate the foreign SIM.
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So I stopped in at an Apple store in London. We logged into a WiFi network, installed a micro-SIM card from a carrier in England, and within about thirty seconds, the phone was unlocked because Verizon had already unlocked the phone at their end before I left the U.S. For the rest of my travels overseas, it worked flawlessly and provided essentially the same options that are available on any other dual-mode or GSM-only device.
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Four different iPhone 4S versions
When I returned to the U.S., I wanted to confirm that my Verizon iPhone could be ported to another carrier, whether CDMA or GSM. This was the intent of Apple when they designed the 4S: every phone is the same and will work on virtually any carrier in the world.
So, I contacted a regional GSM carrier in the Midwest that has roaming agreements with AT&T and T-Mobile. The carrier was kind enough to provide a micro-SIM card for testing in my iPhone. What I found may infuriate customers who have purchased expensive iPhones from one of the three carriers in the U.S. but at some point decide to switch to another provider because of poor coverage, customer service, or pricing.
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Sitting in South Dakota, I replaced my Verizon SIM with the Longlines Wireless chip. The phone indicated that the SIM was not recognized by the phone and required its removal and device reboot. I contacted Verizon global support to confirm my suspicions. They advised that indeed their phones were locked as against any other carrier in the United States, and that the term “unlock” only applied to overseas carriers.
I next inserted my UK SIM card into the phone, and it logged into AT&T without issue, which confirmed what Verizon told me. I then interviewed managers at three different Apple corporate stores to confirm issues with regard to unlocking.
What consumers need to understand is that there are actually four different versions of the iPhone 4S: Verizon, Sprint, AT&T, and Apple. Only the Apple phone, available from their stores or on-line, is fully unlocked and can be used on any carrier outside the United States. The other phones are permanently locked and cannot ever be used on another carrier in the U.S. Even if you spend $800 for an unlocked phone as I did and dedicate it to a single U.S. carrier, you are locked into that carrier forever if you want to keep using the iPhone. Neither Apple or the other carriers will fully unlock your carrier phone.
Best Answer
There are two different iPhone 5s models sold in the US -- the CDMA model for Verizon or Sprint (which also supports GSM connectivity for international use) and the GSM model. The unlocked phone is the GSM model, so it won't support CDMA and won't talk to Verizon's network (except perhaps for data (not voice) on 4G/LTE, where Verizon uses GSM like the rest of the world). So you can buy an unlocked phone and put a Verizon SIM in it, but it will at best only work with Verizon for data, and only in areas where they have 4G coverage.
Just look at the bottom of the page you link to in your question, where it says: