Add span.kind=server to aws.lambda span

This aligns the tracer-created span with the extension's behavior,
enabling server-based features in Datadog APM for Lambda functions.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

Author: jchrostek-dd

src/trace/listener.spec.ts

@@ -113,6 +113,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",
@@ -153,6 +154,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",
@@ -186,6 +188,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",
@@ -227,6 +230,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",
@@ -258,6 +262,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "alias",
@@ -287,6 +292,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "1",
@@ -361,6 +367,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-Lambda",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",
@@ -428,6 +435,7 @@ describe("TraceListener", () => {
         resource: "my-Lambda",
         service: "my-custom-service",
         tags: {
+          "span.kind": "server",
           cold_start: "true",
           function_arn: "arn:aws:lambda:us-east-1:123456789101:function:my-lambda",
           function_version: "$LATEST",

src/trace/listener.ts

@@ -307,6 +307,7 @@ export class TraceListener {
       const functionArn = (this.context.invokedFunctionArn ?? "").toLowerCase();
       const tk = functionArn.split(":");
       options.tags = {
+        "span.kind": "server",
         cold_start: String(didFunctionColdStart()).toLowerCase(),
         function_arn: tk.length > 7 ? tk.slice(0, 7).join(":") : functionArn,
         function_version: tk.length > 7 ? tk[7] : "$LATEST",